Geometry and shapes questions, answered

Sector Exposure Calculator NZ 2026/27

How do I find my total sector exposure?

Add the sector allocations of every fund you hold, weighted by the amount in each. On the worked example three funds holding $35,000.00 between them produce a portfolio that is 39.14% technology, even though the largest of the three funds is only 26% technology on its own. Nothing reports this figure, because each fund is presented separately.

How much of one sector is too much?

There is no threshold, and the useful question is whether the figure was chosen. On the worked example technology is 39.14% of the portfolio and the top three sectors are 65.00% between them. Neither number is wrong in itself. What makes it a problem is arriving there by holding a broad fund and a sector fund without adding them up.

What is the effective number of sectors?

The number of equally sized sectors that would give the same concentration as your actual weights. It is calculated from a Herfindahl index, the sum of the squared weights. On the worked example eight sectors behave like 4.65, because technology alone carries nearly two fifths of the money and the smallest sectors carry almost none.

What happens if my largest sector falls?

The portfolio moves by the fall multiplied by that sector's weight. On the worked example a 30% fall in technology alone, with every other sector flat, costs 11.74% of the portfolio, or $4,110.00 of $35,000.00. A 50% fall costs $6,850.00. Sector falls of that size happen without the wider market doing anything unusual.

Do different funds use the same sector names?

Not consistently. Sector classification schemes range from eight to eleven categories, and providers differ on where they place things like listed property, telecommunications and consumer companies. Group each factsheet's categories into the eight used here rather than omitting anything, and accept that the result is approximate at the boundaries while being accurate on the broad picture.

Does a broad index fund solve sector concentration?

It reflects the market's sector weights, which is not the same as spreading evenly across them. A market-weighted global fund carries whatever sector concentration the market currently has, and that has been substantial at various times. Holding one broad fund removes the risk of accidentally doubling up; it does not produce an even sector split, and it is not trying to.

How does this differ from checking my country exposure?

They frequently point at the same problem from different angles. A portfolio heavy in one country is often heavy in that country's dominant sectors, and a portfolio heavy in one sector is often concentrated in the country where that sector is listed. Running both is worthwhile, because a position that looks moderate on one measure can look extreme on the other.

What should I do if one sector dominates?

Usually redirect new contributions rather than sell, since that costs nothing and realises nothing. Decide first whether the concentration was intended, because a deliberate sector position is a legitimate strategy and only becomes a problem when it was never chosen. Adding a fourth fund to correct it is rarely the answer, since that is generally how portfolios end up in this state.

NZ Property Sale Tax Triggers Calculator 2026

Is there a capital gains tax in New Zealand?

New Zealand has no general capital gains tax, but it has an intention-based tax system for land and several specific provisions in the Income Tax Act 2007 that catch various property sale scenarios. If you bought with the intention of resale, sold within 2 years (bright-line), are a dealer/developer/builder, or your sale involves subdivision or rezoning, the gain is taxable as ordinary income. Many people assume 'no CGT means no tax' and are surprised to get a tax bill from a sale. Run your situation through each of the land sale provisions before assuming you're safe.

What is the 'purpose of resale' test in section CB 6?

Section CB 6 catches any land acquired with an intention or purpose of resale, regardless of how long you hold it. This is the broadest provision - there's no time limit. If at the time of acquiring the land you had an intention to resell (even a subordinate one), the gain on sale is fully taxable. The intention is assessed at the time of acquisition based on contemporaneous evidence (correspondence, what you told others, circumstances of the purchase). A pure investment intention (rent and hold indefinitely) usually escapes section CB 6 - but if you told the real estate agent 'I'm going to flip this', you've created evidence that will follow you.

Who is a 'property dealer' under section CB 7?

Anyone who deals in land as a business. This isn't limited to people with a 'real estate dealer' title - it's based on the actual pattern of activity. Buying, renovating, and selling 3+ properties in quick succession can make you a dealer. Once you're classified as a dealer, ALL land sold is taxable regardless of holding period or bright-line status. Dealers also cannot use the main home exclusion. Section CB 7 has severe 'tainting' effects - personal sales can be tainted by your business. If you're unsure whether your activity counts, get tax advice before making the next purchase.

What is the 10-year builder rule?

Section CB 9 catches sales of land by builders (or people associated with builders) within 10 years of the building work. If you or your company erected or contracted the erection of a building on the land, selling within 10 years of that work makes the gain taxable. This applies even if the building work was on a neighbouring property - the tainting can be wide. 'Builder' is defined broadly - it includes companies and their shareholders, not just registered LBPs. If you've done any construction work in connection with the land, check CB 9 before selling.

How do all these rules interact?

Only one rule needs to apply to make the gain taxable. Each provision is tested independently. The bright-line test (s CB 6A) is the most automatic - it applies to residential property regardless of intention. The other rules (purpose of resale, dealer, developer, builder, subdivision, rezoning) each have their own tests. If ANY of them catch your sale, you're taxable. Exclusions (main home, farmland, business premises) only apply to specific provisions and not others. This calculator tests each rule independently so you can see which specific provision is the issue.

Is Your Area Growing or Shrinking?

How fast is New Zealand's population growing?

Summed across the Stats NZ population grid, the national figure rose from about 5,079,605 in 2022 to about 5,325,333 in 2025, roughly 4.8 per cent over three years. The national figure is the least interesting version of the number: growth is wildly uneven between areas, and the local figure is the one that moves rents, school rolls and shop viability.

Which part of New Zealand is growing fastest?

Measured on the population within 10 kilometres of each published area, with small catchments excluded, the fastest growth from 2022 to 2025 is around Burnham Camp in Canterbury, up about 20.5 per cent, from 32,479 to 39,142 people. The slowest is around Thames North in Waikato, at -1.7 per cent over the same three years.

Are these Census counts?

No. They are modelled estimates: Stats NZ distributes its population estimates across a grid of one kilometre cells, and values are fractional. They are reliable summed over a catchment and progressively less reliable the smaller the area, which is why figures built on too few cells are not shown at all on this site.

Why does local population growth matter to ordinary money decisions?

Because it is the demand side of almost everything local. Rents and prices in a growing catchment sit on rising demand; a shrinking catchment does the opposite quietly. School zones tighten with growth. A business's customers within reach grow or shrink with it. It is the number working in the background of decisions people usually make on feel.

Rectangle Calculator

How do I calculate the area of a rectangle?

Multiply the length by the width. The result is in square units, so a rectangle 5 metres long and 3 metres wide has an area of 15 square metres.

How do I calculate the perimeter of a rectangle?

Add the length and the width and multiply by two, because a rectangle has two pairs of equal sides. A rectangle 5 metres by 3 metres has a perimeter of 2 times 8, which is 16 metres.

How do I find the diagonal of a rectangle?

Use the Pythagorean theorem. The diagonal is the square root of the length squared plus the width squared, because the diagonal is the hypotenuse of a right triangle formed by two sides. A 5 by 3 rectangle has a diagonal of the square root of 34, about 5.83.

What units does the rectangle calculator use?

You choose the unit, such as millimetres, centimetres, metres, kilometres, inches or feet. The length, width, perimeter and diagonal share that unit, and the area is given in the matching square unit.

Triangle Area Calculator (3 Sides)

How do you find the area of a triangle from three sides?

Use Heron's formula. First calculate the semi-perimeter: s = (a + b + c) / 2. Then compute the area: Area = sqrt(s x (s - a) x (s - b) x (s - c)). For example, a triangle with sides 5, 6, and 7 has s = 9 and area = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.70 square units. The formula works for any valid triangle regardless of whether the angles are known.

What is a valid triangle? Why might the calculator show an error?

Three lengths a, b, and c form a valid triangle only if the sum of any two sides is greater than the third side. This is called the triangle inequality: a + b > c, a + c > b, and b + c > a. If any condition fails, no triangle can be formed. For example, sides 1, 2, and 10 are not a valid triangle because 1 + 2 = 3 is less than 10. If you enter such values the calculator will display an error rather than a result.

What is the difference between an equilateral, isosceles, and scalene triangle?

An equilateral triangle has all three sides equal in length. An isosceles triangle has exactly two sides equal. A scalene triangle has all three sides of different lengths. The triangle area calculator identifies which type your triangle is based on the side lengths you enter. All three types can be solved using Heron's formula.

45-45-90 Triangle Calculator

What are the side ratios of a 45-45-90 triangle?

In a 45-45-90 triangle the two legs are equal in length and the hypotenuse is exactly the leg length multiplied by the square root of 2 (approximately 1.41421). So if each leg is 1, the hypotenuse is √2. If each leg is 5, the hypotenuse is 5√2 which is approximately 7.071. This ratio comes directly from Pythagoras: a² + a² = c², so 2a² = c², giving c = a√2.

How do you find the leg of a 45-45-90 triangle from the hypotenuse?

Divide the hypotenuse by the square root of 2. For example, if the hypotenuse is 10, each leg = 10 / √2 = 10√2 / 2 = 5√2, which is approximately 7.071. You can also multiply the hypotenuse by √2 / 2, which gives the same result. This is the reverse of the standard ratio c = a√2.

What is the area formula for a 45-45-90 triangle?

Because the two legs are equal and they form the base and height of the right angle, the area is simply leg² / 2. For example, if each leg is 6, the area is 6² / 2 = 36 / 2 = 18 square units. If you only know the hypotenuse, first find the leg (hypotenuse / √2) then apply the same formula.

AAA Triangle Calculator

Can you find the sides of a triangle from angles alone (AAA)?

Not without at least one side length. Three angles (AAA) define the shape and proportions of a triangle but not its size. Any triangle with the same three angles is similar to yours but could be any scale. Once you know one side, the Law of Sines determines all remaining sides: a/sin(A) = b/sin(B) = c/sin(C). This ratio is called the circumscribed circle diameter (2R). Without a side, you can still classify the triangle (acute, right, or obtuse; scalene, isosceles, or equilateral) and find the ratios between the sides.

What does AAA mean in triangle geometry?

AAA stands for Angle-Angle-Angle, meaning all three interior angles of a triangle are known. In geometry, AAA is a triangle similarity criterion: if two triangles have the same three angle measures, they are similar (same shape, proportional sides). AAA is not a congruence criterion because it does not fix the size of the triangle. By contrast, SSS (three sides), SAS (two sides and included angle), and ASA or AAS (two angles and a side) are all congruence criteria that fully determine a unique triangle.

How do I classify a triangle from its angles?

A triangle is classified by its largest angle. If all angles are less than 90°, it is acute. If one angle equals exactly 90°, it is a right triangle. If one angle is greater than 90°, it is obtuse. By side lengths (or equivalently by angle equality): if all three angles are equal (each 60°) the triangle is equilateral; if exactly two angles are equal it is isosceles; if all three differ it is scalene. Note that an equilateral triangle is always also equiangular, and a right triangle can be isosceles (the 45-45-90 triangle) but never equilateral.

AAS Triangle Calculator

What is AAS in geometry?

AAS stands for Angle-Angle-Side. It is a method of defining a triangle by giving two of its interior angles and a side that is not between those two angles (a non-included side). Because the three interior angles of any triangle must sum to 180 degrees, knowing two angles immediately gives you the third. From there, the Law of Sines lets you find all remaining sides. AAS always produces a unique triangle (unlike SSA, which can be ambiguous).

How do you solve an AAS triangle?

To solve an AAS triangle: (1) Find the third angle using C = 180 degrees minus A minus B. (2) Use the Law of Sines: a / sin(A) = b / sin(B) = c / sin(C), where a is the side opposite angle A. If side a and angles A and B are known, then b = a times sin(B) / sin(A) and c = a times sin(C) / sin(A). (3) Calculate the perimeter as a + b + c. (4) Calculate the area using Area = (a times b times sin(C)) / 2.

What is the difference between AAS and ASA?

In ASA (Angle-Side-Angle) the known side is located between the two known angles (the included side). In AAS (Angle-Angle-Side) the known side is opposite one of the known angles, not between them. Both configurations always produce a unique triangle, but the label matters for labelling diagrams consistently. Mathematically both are solved with the Law of Sines once the third angle is calculated.

ABC Triangle Calculator (SSS)

How do you find the angles of a triangle when you know all three sides?

Use the law of cosines. For a triangle with sides a, b, and c, angle A (opposite side a) is found with: A = arccos((b² + c² - a²) / (2bc)). Angles B and C are found similarly by cycling the letters. Once you have two angles, the third is simply 180° minus the sum of the other two.

What is Heron's formula for triangle area?

Heron's formula calculates the area of a triangle from its three side lengths without needing the height. First compute the semi-perimeter: s = (a + b + c) / 2. Then the area is: Area = sqrt(s × (s - a) × (s - b) × (s - c)). For example, a triangle with sides 5, 6, and 7 has s = 9 and area = sqrt(9 × 4 × 3 × 2) = sqrt(216) ≈ 14.70 square units.

When do three side lengths NOT form a valid triangle?

Three side lengths form a valid triangle only if each side is strictly less than the sum of the other two. This is called the triangle inequality. For example, sides 1, 2, and 5 do not form a triangle because 1 + 2 = 3, which is less than 5. The calculator will warn you if the entered sides cannot form a real triangle.

Acute Triangle Calculator

What makes a triangle acute?

A triangle is acute when all three of its interior angles are strictly less than 90 degrees. Equivalently, for a triangle with sides a, b, and c (where c is the longest side), it is acute if and only if a squared plus b squared is greater than c squared. This is the reverse of the Pythagorean condition: in a right triangle a squared plus b squared equals c squared, and in an obtuse triangle a squared plus b squared is less than c squared. Because angles in any triangle add to 180 degrees, having all three angles under 90 degrees means no single angle dominates the shape.

How do you find the angles of an acute triangle from its sides?

Use the law of cosines: for angle A opposite side a, the formula is cos(A) = (b squared + c squared minus a squared) divided by (2bc). Rearranging gives A = arccos((b squared + c squared minus a squared) / (2bc)). Apply the same formula cyclically to find angles B and C. Once two angles are known, the third is 180 minus A minus B degrees. If all three resulting angles are less than 90 degrees and greater than zero, the triangle is acute. If any cosine value is negative, the corresponding angle exceeds 90 degrees and the triangle is obtuse.

Can an equilateral triangle be acute?

Yes. An equilateral triangle has all three angles equal to exactly 60 degrees, which are all less than 90 degrees, so it is always an acute triangle. It is in fact the most symmetric acute triangle. Any isosceles triangle with the apex angle less than 90 degrees is also acute, provided both base angles (which are equal) are also less than 90 degrees. For example, an isosceles triangle with angles 70, 70, and 40 degrees is acute.

Angle Between Two Vectors Calculator

What formula is used to find the angle between two vectors?

The angle theta between two vectors A and B is found using the dot product formula: cos(theta) = (A dot B) / (|A| times |B|), where A dot B is the dot product of the vectors and |A| and |B| are their magnitudes (lengths). Solving for theta gives: theta = arccos((A dot B) / (|A| times |B|)). This formula works for both 2D and 3D vectors and always returns an angle between 0 and 180 degrees.

What does it mean if the angle between two vectors is 90 degrees?

If the angle between two vectors is exactly 90 degrees, the vectors are perpendicular (also called orthogonal). This happens when their dot product equals zero. For example, the vectors (1, 0) and (0, 1) are perpendicular because 1 times 0 plus 0 times 1 equals 0. Perpendicular vectors are important in physics, engineering, and mathematics because forces or components at right angles to each other do not influence one another along the perpendicular direction.

What is the difference between the angle in degrees and radians?

Degrees and radians are two units for measuring angles. A full circle is 360 degrees or 2 pi radians. To convert degrees to radians, multiply by pi divided by 180. To convert radians to degrees, multiply by 180 divided by pi. For example, 90 degrees equals pi divided by 2 radians (approximately 1.5708 radians). Scientific calculators and most programming languages use radians by default, while degrees are more familiar for everyday use.

Angle Cut Calculator - Miter and Bevel Angles for Woodworking

How do I calculate the miter angle for a corner?

For a simple flat join (such as skirting board or picture frame), the miter angle you set on the saw for each piece is 90 degrees minus half the corner angle. A standard 90-degree corner requires each piece to be cut at 45 degrees. A 135-degree bay-window corner requires each piece to be cut at 22.5 degrees, and a 120-degree corner requires 30 degrees. The formula is: miter angle = 90 - (corner angle / 2). For a compound cut on a sloped or angled surface, both a miter angle (horizontal blade rotation) and a bevel angle (blade tilt) are needed, and the formulas involve trigonometry.

What is the difference between a miter cut and a bevel cut?

A miter cut is a cut made at an angle across the face (width) of a piece of timber, with the saw blade kept vertical. You set the miter angle by rotating the saw table or fence left or right. A bevel cut is a cut made at an angle through the thickness of the piece, with the saw blade tilted away from vertical. A compound cut uses both: the table is rotated (miter angle) and the blade is tilted (bevel angle). Compound cuts are needed when joining pieces that meet at a corner where one or both surfaces are not flat or vertical, such as crown moulding or sloped roof trim.

What miter angle do I use for a six-sided hexagonal frame?

A regular hexagon has interior corner angles of 120 degrees. To join six pieces into a hexagonal frame, each end of each piece must be cut at half the exterior supplement angle. The interior angle is 120 degrees, so the miter cut on each end is 30 degrees (because 180 degrees minus 120 degrees equals 60 degrees, and half of that is 30 degrees). You can also calculate this as 90 degrees minus (interior angle / 2), which gives 90 minus 60 equals 30 degrees. For a square (4 sides, 90-degree corners), this gives 90 minus 45 equals 45 degrees.

Angle of Depression Calculator

What is the formula for the angle of depression?

theta = arctan(vertical height / horizontal distance). The tangent of the angle is the height divided by the horizontal distance.

What is the difference between elevation and depression?

Elevation is the angle looking up from the horizontal, depression is the angle looking down. The arithmetic is identical; only the direction differs.

What is the line of sight value?

The straight distance to the point, the hypotenuse, equal to sqrt(distance squared plus height squared).

Angle of Elevation Calculator

What is the formula for the angle of elevation?

theta = arctan(vertical height / horizontal distance). The tangent of the angle is the height divided by the horizontal distance.

What is the difference between elevation and depression?

Elevation is the angle looking up from the horizontal, depression is the angle looking down. The arithmetic is identical; only the direction differs.

What is the line of sight value?

The straight distance to the point, the hypotenuse, equal to sqrt(distance squared plus height squared).

Angle of Impact Calculator

What formula is used to calculate the angle of impact?

The standard formula is: angle of impact = arcsin(width / length), where width is the minor axis (the short dimension) of the elliptical stain and length is the major axis (the long dimension). This formula is derived from the geometry of a sphere striking a flat surface at an angle: at 90 degrees (perpendicular) the stain is circular (width equals length), and at shallower angles the stain becomes more elongated. The result is in degrees.

Why does a low angle produce a more elongated stain?

A droplet travelling at a low angle to a surface strikes it with a large component of velocity parallel to the surface and a small component perpendicular to it. The parallel velocity stretches the stain along the direction of travel, creating a narrow ellipse. At 90 degrees (directly perpendicular) there is no parallel component and the stain is circular. The ratio of width to length captures this geometry, and arcsin converts that ratio back into the original angle of impact.

What are the limits of the angle of impact formula?

The formula assumes a smooth, level, hard surface and a spherical droplet. Textured surfaces, curved surfaces, or high-velocity impacts that cause spatter tails can distort the stain shape and introduce error. Measurement accuracy matters: even small errors in measuring width are amplified at very low angles (below about 10 degrees) because arcsin is sensitive near zero. For forensic work, multiple stains are measured and averaged to improve reliability.

Angle of Incidence Calculator

What is the angle of incidence?

The angle of incidence is the angle between an incoming ray of light (or other wave) and the normal (a line perpendicular to the surface) at the point where the ray strikes the surface. It is always measured from the normal, not from the surface itself. When a ray hits a boundary between two media at 0 degrees to the normal, it passes straight through without bending. As the angle increases, the ray bends more on entering the new medium.

How does Snell's Law relate angle of incidence to angle of refraction?

Snell's Law states that n1 times sin(theta1) equals n2 times sin(theta2), where n1 is the refractive index of the first medium, theta1 is the angle of incidence, n2 is the refractive index of the second medium, and theta2 is the angle of refraction. All angles are measured from the normal to the surface. If the second medium is denser (higher n), the refracted ray bends toward the normal and the angle of refraction is smaller than the angle of incidence.

What is the law of reflection for the angle of incidence?

The law of reflection states that the angle of incidence equals the angle of reflection. Both angles are measured from the normal to the reflecting surface. This law holds for all smooth (specular) surfaces, including mirrors and still water. The incident ray, the reflected ray, and the normal all lie in the same plane. This is distinct from refraction, which involves a ray passing into a second medium and bending according to Snell's Law.

Angle of Refraction Calculator

What is the angle of refraction?

The angle of refraction is the angle between a refracted ray of light and the normal (a line perpendicular to the boundary surface) at the point where the ray enters a new medium. When light travels from one transparent medium to another with a different refractive index, it changes direction. The angle of refraction depends on the angle of incidence and the ratio of the refractive indices of the two media, as described by Snell's Law: n1 sin(theta1) = n2 sin(theta2).

What happens when the angle of incidence exceeds the critical angle?

When light travels from a denser medium to a less dense medium (for example from glass into air) and the angle of incidence exceeds the critical angle, total internal reflection occurs. No light is transmitted into the second medium; instead, all light reflects back into the first medium. The critical angle is calculated as arcsin(n2 / n1) where n1 is the denser medium. This principle is used in optical fibres to carry light signals over long distances with very low loss.

What are the refractive indices of common materials?

Common refractive indices include: vacuum = 1.000 (exact), air = 1.000 (approximately), water at 20 degrees Celsius = 1.333, crown glass = 1.520, flint glass = 1.620, diamond = 2.417, and acrylic (PMMA) = 1.490. The refractive index of a material is the ratio of the speed of light in a vacuum to the speed of light in that material. A higher refractive index means light travels more slowly through that material and bends more when entering from a less dense medium.

Angle of Repose Calculator

What is the angle of repose?

The angle of repose is the steepest angle at which a pile of granular material (such as sand, gravel, or soil) can sit without sliding. At this angle, the gravitational force component pulling particles down the slope exactly equals the maximum frictional force holding them in place. It is numerically equal to the arctangent of the coefficient of static friction between the particles: phi = arctan(mu). For example, dry sand has a coefficient of static friction of about 0.58, giving an angle of repose of roughly 30 degrees.

How do you calculate the angle of repose from a pile?

If you can measure the height (h) and the base diameter (d) of a conical pile, the angle of repose is calculated using phi = arctan(2h / d), where 2h/d represents the rise over the half-base (the slope ratio). For example, a pile 1.2 m high with a base diameter of 4 m gives phi = arctan(2 x 1.2 / 4) = arctan(0.6) = approximately 31 degrees.

What are typical angles of repose for common materials?

Typical angles of repose are: dry sand 28 to 34 degrees (mu approx 0.53 to 0.67); gravel 35 to 45 degrees (mu approx 0.70 to 1.00); wet sand 45 degrees (mu approx 1.00); dry soil 30 to 45 degrees; coal 38 degrees (mu approx 0.78); wheat 27 degrees (mu approx 0.51); wood chips 45 to 50 degrees. Materials with rougher or more angular particles generally have higher angles of repose.

Angle of Twist Calculator

What is the angle of twist formula?

The angle of twist is given by phi = TL / (GJ), where T is the applied torque (N.m), L is the shaft length (m), G is the shear modulus of the material (Pa), and J is the polar moment of inertia of the cross-section (m⁴). For a solid circular shaft of diameter d, J = pi * d⁴ / 32. For a hollow shaft with outer diameter d_o and inner diameter d_i, J = pi * (d_o⁴ - d_i⁴) / 32. The result phi is in radians; multiply by 180/pi to convert to degrees.

What is the shear modulus for common engineering materials?

The shear modulus (also called modulus of rigidity, G) measures a material's resistance to shear deformation. Typical values: Structural steel: 80 GPa; Stainless steel: 77 GPa; Aluminium alloy: 26 GPa; Copper: 45 GPa; Brass: 37 GPa; Cast iron: 41 GPa; Titanium alloy: 44 GPa. These values vary slightly between alloy grades and should be confirmed with material data sheets for critical engineering applications.

What is the maximum allowable angle of twist for a shaft?

There is no single universal limit; it depends on the application. A common design rule for general engineering shafts is to limit the angle of twist to 0.25 to 1 degree per metre of shaft length. For precision machinery, limits as tight as 0.1 degrees per metre may apply. Transmission shafts are often limited to 1 degree per metre. Where the application is sensitive to angular displacement (such as machine tool spindles or precision drives), the designer should check the maximum permissible twist angle specified for that system.

Aortic Valve Area Calculator (Continuity Equation)

What is the continuity equation for aortic valve area?

The continuity equation states that blood flow is conserved across the left ventricular outflow tract (LVOT) and the aortic valve. AVA = (LVOT area x LVOT VTI) / AV VTI. LVOT area is calculated from the LVOT diameter as pi x (diameter/2)^2. VTI (velocity-time integral) is measured by pulsed Doppler at the LVOT and continuous-wave Doppler at the aortic valve. This is the recommended method in current clinical guidelines for assessing aortic stenosis severity.

What AVA indicates severe aortic stenosis?

According to ACC/AHA and ESC guidelines, severe aortic stenosis is defined as an AVA of less than 1.0 cm^2 (or indexed AVA less than 0.6 cm^2/m^2 of body surface area). Moderate stenosis is 1.0 to 1.5 cm^2, and mild stenosis is greater than 1.5 cm^2. Very severe stenosis is sometimes defined as AVA less than 0.6 cm^2. The AVA is used alongside mean gradient (severe at over 40 mmHg) and peak velocity (severe at over 4 m/s) to classify stenosis. Discordant findings between these parameters require additional evaluation.

What is the difference between AVA and indexed AVA?

Aortic valve area (AVA) is the absolute opening area of the valve in cm^2. Indexed AVA (AVAi) adjusts for body size by dividing AVA by the body surface area (BSA) in m^2. AVAi is particularly useful in patients with small or large body habitus because a small absolute AVA may represent less severe stenosis in a very small patient, and a larger AVA may still represent significant stenosis in a very large patient. The threshold for severe stenosis by AVAi is less than 0.6 cm^2/m^2.

Aperture Area Calculator

How do you calculate the area of a circular aperture?

The area of a circular aperture is calculated using A = π × r², where r is the radius (half the diameter). If you know the diameter D, the formula becomes A = π × (D/2)². For example, an aperture with a diameter of 50 mm has a radius of 25 mm, giving an area of π × 25² = approximately 1,963.50 mm².

What is the relationship between f-number and aperture area?

The f-number (f-stop) of a lens is the ratio of focal length to effective aperture diameter: f/# = focal length ÷ aperture diameter. Rearranging, aperture diameter = focal length ÷ f-number. For a 200 mm focal length lens at f/4, the effective aperture diameter is 200 ÷ 4 = 50 mm, giving an area of approximately 1,963.50 mm². Each full f-stop reduces the aperture area by half, because the f-number increases by a factor of √2 (e.g., f/4 to f/5.6).

Why does aperture area matter for telescopes and cameras?

Aperture area determines how much light a lens or mirror collects. Because light collection is proportional to area (not diameter), doubling the aperture diameter quadruples the light gathered. In astronomy, a larger aperture resolves finer detail and detects fainter objects. In photography, a wider effective aperture (lower f-number) lets in more light, enabling faster shutter speeds or lower ISO settings in dim conditions. For a given sensor or film size, aperture area also governs depth of field.

Arc Length and Sector Area Calculator

What is arc length?

Arc length is the distance along the curved edge of a circular sector, measured along the circumference. It equals the radius multiplied by the central angle in radians. For a full circle (360 degrees or 2pi radians) the arc length equals the full circumference 2 pi r. For a quarter circle (90 degrees) the arc is pi r divided by 2.

What is a sector?

A sector is the pie-slice-shaped region bounded by two radii and an arc of a circle. Its area equals one half times the radius squared times the central angle in radians. Equivalently, the sector area is the fraction (angle / 360) of the full circle area pi r squared when the angle is in degrees.

What is a chord?

A chord is the straight line segment connecting the two endpoints of an arc. Its length is 2r times sin(theta divided by 2), where theta is the central angle in radians. For a semicircle (180 degrees) the chord equals the diameter 2r. For small angles the chord approaches the arc length.

Area Calculator (Multi-Shape)

How do you calculate the area of a circle?

The area of a circle is calculated using the formula A = pi x r squared, where r is the radius (half the diameter) and pi is approximately 3.14159. For example, a circle with radius 5 m has an area of 3.14159 x 25 = 78.5398 square metres. If you know the diameter instead of the radius, divide it by 2 first.

What is the formula for the area of a trapezoid?

The area of a trapezoid (also called a trapezium) is A = ((a + b) / 2) x h, where a and b are the lengths of the two parallel sides and h is the perpendicular height between them. For example, a trapezoid with parallel sides of 6 m and 10 m and a height of 4 m has an area of ((6 + 10) / 2) x 4 = 8 x 4 = 32 square metres.

How is the area of an ellipse calculated?

The area of an ellipse is A = pi x a x b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). For example, an ellipse with semi-axes of 6 m and 4 m has an area of 3.14159 x 6 x 4 = 75.3982 square metres. When both axes are equal the formula simplifies to the circle formula A = pi x r squared.

Area Calculator

How do you calculate the area of a circle?

The area of a circle is pi times the radius squared, written as A = pi r squared. If you know the diameter instead of the radius, divide by two first. The circumference (perimeter) is 2 times pi times r.

What is the area formula for a trapezoid?

A trapezoid has two parallel sides called bases (a and b) and a height (h) measured perpendicularly between them. The area is half the sum of the parallel sides multiplied by the height: A = (a + b) / 2 times h.

How is the perimeter of an ellipse calculated?

There is no simple exact formula for the perimeter (circumference) of an ellipse. This calculator uses Ramanujan's second approximation: P is approximately pi(a+b)[1 + 3h/(10 + sqrt(4 - 3h))], where h = (a - b)^2 / (a + b)^2 and a and b are the semi-major and semi-minor axes. It is accurate to better than about 0.01% for virtually all ellipses.

Area of a Circle Calculator

What is the formula for the area of a circle?

The area of a circle is calculated using the formula A = pi times r squared, where r is the radius (the distance from the centre to the edge) and pi is approximately 3.14159. If you know the diameter (d) instead of the radius, use A = pi times (d divided by 2) squared, or equivalently A = pi times d squared divided by 4.

How do I find the area of a circle from its diameter?

If you know the diameter, first halve it to get the radius (r = d / 2), then apply the formula A = pi times r squared. For example, a circle with a diameter of 10 cm has a radius of 5 cm, giving an area of pi times 25 = 78.54 cm squared. You can also use the shortcut formula A = pi times d squared divided by 4.

What units does the area of a circle use?

The area is always expressed in square units. If you enter the radius in centimetres (cm), the area will be in square centimetres (cm squared). If you use metres (m), the result is in square metres (m squared). The circumference uses the same linear unit as the radius (for example, cm if the radius is in cm). Use the unit selector to label your results correctly.

Area of a Rectangle Calculator

What is the formula for the area of a rectangle?

The area of a rectangle is calculated by multiplying its length by its width: Area = length x width. For example, a rectangle that is 10 metres long and 5 metres wide has an area of 10 x 5 = 50 square metres. The result is always expressed in square units (such as m^2, cm^2, or ft^2) because you are multiplying two linear measurements together.

How do I find the perimeter of a rectangle?

The perimeter of a rectangle is the total distance around its outside edge. The formula is Perimeter = 2 x (length + width). For a rectangle that is 10 m long and 5 m wide, the perimeter is 2 x (10 + 5) = 2 x 15 = 30 metres. You can also think of it as adding all four sides: length + width + length + width.

How do I calculate the diagonal of a rectangle?

The diagonal of a rectangle is found using Pythagoras's theorem, because the diagonal forms the hypotenuse of a right-angled triangle with the length and width as the other two sides. The formula is Diagonal = sqrt(length^2 + width^2). For a 10 m x 5 m rectangle, the diagonal is sqrt(10^2 + 5^2) = sqrt(100 + 25) = sqrt(125) = 11.18 metres (to 2 decimal places).

Area of a Regular Polygon Calculator

What is the formula for the area of a regular polygon?

For a regular polygon with n sides each of length s, the area is A = (n x s^2) / (4 x tan(pi/n)). Equivalently, if you know the apothem a (the perpendicular distance from the centre to a side), the area is A = (1/2) x perimeter x a = (1/2) x (n x s) x a. If you know the circumradius R (distance from centre to a vertex), the area is A = (1/2) x n x R^2 x sin(2pi/n). All three formulas give the same result when the same polygon is described.

What is the difference between the apothem and the circumradius of a polygon?

The apothem (also called the inradius) is the perpendicular distance from the centre of a regular polygon to the midpoint of any side. It is the radius of the inscribed circle. The circumradius is the distance from the centre to any vertex (corner), and is the radius of the circumscribed circle. For a given polygon, the circumradius is always larger than the apothem. For a regular hexagon with side length s, the apothem is s x sqrt(3)/2 and the circumradius equals s.

How do I find the area of a regular hexagon?

For a regular hexagon (6 sides) with side length s, the area formula simplifies to A = (3 x sqrt(3) / 2) x s^2. For example, a hexagon with side length 10 cm has area = (3 x 1.73205 / 2) x 100 = 2.59808 x 100 = 259.81 cm^2 (rounded to 2 decimal places). This matches the general formula A = (n x s^2) / (4 x tan(pi/n)) with n = 6.

Area of a Right Triangle Calculator

What is the formula for the area of a right triangle?

The area of a right triangle is calculated using the formula Area = ½ × base × height, where the base and height are the two legs that meet at the right angle (90°). For example, a right triangle with a base of 6 and a height of 8 has an area of ½ × 6 × 8 = 24 square units.

How do I find the hypotenuse of a right triangle?

The hypotenuse is the longest side of a right triangle, opposite the right angle. You find it using the Pythagorean theorem: hypotenuse = √(base² + height²). For a triangle with base 6 and height 8, the hypotenuse is √(36 + 64) = √100 = 10.

What are the angles in a right triangle?

Every right triangle has one angle of exactly 90° (the right angle) and two acute angles that together sum to 90°. The angle adjacent to the base is calculated as arctan(height ÷ base), and the angle adjacent to the height is arctan(base ÷ height). All three angles always add up to 180°.

Area of a Square Calculator

What is the formula for the area of a square?

The area of a square is calculated by multiplying the side length by itself: Area = side x side, or Area = s². For example, a square with a side length of 5 cm has an area of 25 cm². The unit of the result is always the square of the unit used for the side (so cm gives cm², m gives m², and so on).

How do you find the perimeter of a square?

The perimeter of a square is the total distance around all four sides: Perimeter = 4 x side length. For a square with side 5 cm, the perimeter is 4 x 5 = 20 cm. Because all four sides of a square are equal, you simply multiply the side length by 4.

How do you calculate the diagonal of a square?

The diagonal of a square is found using the Pythagorean theorem. Because a square has four equal sides and four right angles, the diagonal d = s x sqrt(2), where s is the side length. For a square with side 5 cm, the diagonal is 5 x 1.41421 = approximately 7.071 cm. The diagonal splits the square into two right-angled triangles where the diagonal is the hypotenuse.

Area of a Trapezoid Calculator

What is the formula for the area of a trapezoid?

The area of a trapezoid is calculated using A = (a + b) / 2 x h, where a and b are the lengths of the two parallel sides (called the bases) and h is the perpendicular height between those bases. This is equivalent to multiplying the average of the two bases by the height. For example, a trapezoid with bases of 6 cm and 10 cm and a height of 4 cm has an area of (6 + 10) / 2 x 4 = 8 x 4 = 32 cm squared.

What is the difference between a trapezoid and a trapezium?

The terms are regional variations of the same shape. In New Zealand, Australia, and the United Kingdom, the shape with one pair of parallel sides is called a trapezium. In the United States and Canada, the same shape is called a trapezoid. On this calculator, the word trapezoid is used in its American sense: a quadrilateral with exactly one pair of parallel sides (called the bases). The area formula A = (a + b) / 2 x h applies regardless of which term you use.

Does it matter which side I enter as base a and which as base b?

No. Because the formula adds the two bases together before dividing by two, the result is identical whether you enter the longer base as a or b. What does matter is that you enter the perpendicular height, which is the vertical distance between the two parallel sides, not the length of either slanting leg. If you have the leg length instead of the height, you can use the Pythagorean theorem to find the perpendicular height first.

Area of a Triangle Calculator

How do you find the area of a triangle?

The simplest way is half the base times the perpendicular height. If you only know the three side lengths, use Heron's formula instead, which works from the sides alone.

What is Heron's formula?

Heron's formula finds the area from the three sides. Work out the semi-perimeter s, half the sum of the sides, then take the square root of s times s minus a, times s minus b, times s minus c.

Do the three sides always form a triangle?

Only if each side is shorter than the sum of the other two, which is the triangle inequality. If not, no triangle exists and the calculator will say so.

Area of a Triangle (Heron's Formula) Calculator

What is Heron's formula for the area of a triangle?

Heron's formula calculates the area of a triangle from its three side lengths (a, b, c) without needing an angle or the height. First compute the semi-perimeter: s = (a + b + c) / 2. Then the area is: Area = sqrt(s x (s - a) x (s - b) x (s - c)). The formula works for any valid triangle, including scalene, isosceles, and equilateral triangles.

When does a set of three sides NOT form a valid triangle?

Three sides form a valid triangle only when each side is shorter than the sum of the other two. This is called the triangle inequality. For example, sides of 1, 2, and 10 do not form a triangle because 1 + 2 = 3, which is less than 10. If this condition is violated, the value under the square root in Heron's formula becomes negative, indicating the sides cannot close into a triangle. This calculator checks the triangle inequality and shows an error if the sides are invalid.

How is Heron's formula different from the base-times-height formula?

The standard formula Area = 0.5 x base x height requires you to know the perpendicular height, which can be difficult to measure directly on an irregular triangle. Heron's formula only needs the three side lengths, making it ideal when you can measure all sides but not the height. Both formulas give the same result. For a right-angled triangle, the two legs serve as base and height, so the standard formula is simpler. For scalene or obtuse triangles where the height falls outside the triangle, Heron's formula is often more convenient.

Area of an Obtuse Triangle Calculator

What makes a triangle obtuse?

A triangle is obtuse when one of its three interior angles is greater than 90 degrees and less than 180 degrees. Because the interior angles of any triangle must sum to exactly 180 degrees, an obtuse triangle can have at most one obtuse angle. The other two angles are always acute (less than 90 degrees). Common examples include a triangle with angles of 120, 35, and 25 degrees.

How do you find the area of an obtuse triangle using base and height?

The area of any triangle, including an obtuse triangle, is Area = 0.5 x base x height, where height is the perpendicular distance from the base to the opposite vertex. For an obtuse triangle the perpendicular height may fall outside the triangle when drawn from a side adjacent to the obtuse angle, but the formula itself is unchanged. For example, a triangle with a base of 8 cm and a perpendicular height of 5 cm has an area of 0.5 x 8 x 5 = 20 square centimetres.

Can you use Heron's formula on an obtuse triangle?

Yes. Heron's formula works for any triangle, including obtuse ones. Given the three side lengths a, b, and c, first compute the semi-perimeter s = (a + b + c) / 2. Then Area = square root of (s x (s - a) x (s - b) x (s - c)). The formula does not require knowing which angle is obtuse; it works purely from the side lengths. The only requirement is that the three lengths form a valid triangle (each side must be less than the sum of the other two).

Area of Quadrilateral Calculator

What is the formula for the area of a quadrilateral?

The formula depends on the type of quadrilateral. For a rectangle or square, Area = length x width. For a parallelogram, Area = base x height (perpendicular height, not slant side). For a trapezoid, Area = 0.5 x (a + b) x h, where a and b are the parallel sides and h is the perpendicular height between them. For a rhombus or kite, Area = 0.5 x d1 x d2, where d1 and d2 are the two diagonals. For a general quadrilateral defined by four corner coordinates, the shoelace formula gives the exact area: Area = 0.5 x |x1(y2 - y4) + x2(y3 - y1) + x3(y4 - y2) + x4(y1 - y3)|.

How do I find the area of an irregular quadrilateral?

For an irregular quadrilateral (one that does not fit a standard shape like a rectangle or trapezoid), use the shoelace formula with the four corner coordinates listed in order (either clockwise or anti-clockwise). The formula is: Area = 0.5 x |(x1*y2 - x2*y1) + (x2*y3 - x3*y2) + (x3*y4 - x4*y3) + (x4*y1 - x1*y4)|. This works for any non-self-intersecting (simple) quadrilateral. Enter the four vertex coordinates into this calculator to get the result instantly.

What is the difference between a parallelogram and a trapezoid?

A parallelogram has two pairs of parallel sides, and opposite sides are equal in length. Its area is base x perpendicular height. A trapezoid (also called a trapezium in New Zealand and the UK) has exactly one pair of parallel sides, called the bases. Its area is 0.5 x (base1 + base2) x height. A rectangle and a rhombus are special cases of a parallelogram. A parallelogram is also a special case of a trapezoid if you use the broader definition (at least one pair of parallel sides).

Area of a Triangle with Coordinates Calculator

What formula is used to find the area of a triangle from coordinates?

The Shoelace formula (also called the coordinate geometry method or cross-product method) is used. For a triangle with vertices (x1, y1), (x2, y2), and (x3, y3), the area is: Area = 0.5 x |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. The absolute value ensures a positive result regardless of the order the vertices are listed.

Does it matter what order I enter the coordinates?

No. The Shoelace formula produces the same area whether you enter the vertices in clockwise or anticlockwise order because the absolute value is taken at the end. However, if all three points are collinear (i.e., they lie on the same straight line), the formula returns zero because no triangle can be formed.

Can coordinates be negative or decimal?

Yes. The formula works with any real numbers, including negative coordinates and decimals. For example, a triangle with vertices at (-3, 1), (2, -4), and (5, 3) is handled correctly. The calculator accepts any numeric input including negative values and fractional coordinates.

Area Under the Curve Calculator

What does area under the curve mean?

The area under the curve is the total region enclosed between the graph of a function and the x-axis over a specified interval. When the function is above the x-axis that area is positive; when it dips below, that region is negative in the signed integral. The total shaded area treats all regions as positive regardless of sign, which is useful for physical problems like total distance travelled or total probability.

How is the area under a curve calculated?

This calculator uses Simpson's rule with 1000 subintervals. Simpson's rule fits a parabola through every pair of adjacent subintervals and sums their areas. It is far more accurate than simple rectangle methods like the Riemann sum for the same number of evaluations, converging at a rate proportional to h to the fourth power, where h is the subinterval width.

What is the average value of a function?

The average value of a function f over [a, b] is the definite integral of f divided by the width of the interval (b minus a). Geometrically it is the height of a rectangle with the same base and same area as the region under the curve. It is used in physics to find average velocity, average force, or average temperature over a time or distance interval.

ASA Triangle Calculator

What is the ASA rule in triangle geometry?

ASA stands for Angle-Side-Angle. It is a triangle congruence rule stating that if you know two angles of a triangle and the side between them (the included side), the triangle is fully determined. You can find the third angle by subtracting the two known angles from 180 degrees, then use the Law of Sines to calculate the remaining two sides.

How do you solve an ASA triangle?

To solve an ASA triangle: (1) Find the third angle: B = 180 - A - C, where A and C are the two known angles. (2) Use the Law of Sines to find the unknown sides: a = b x sin(A) / sin(B) and c = b x sin(C) / sin(B), where b is the known included side opposite angle B. (3) Calculate the area using: Area = (b squared x sin(A) x sin(C)) / (2 x sin(B)). (4) Add all three sides for the perimeter.

What is the included side in an ASA triangle?

The included side in an ASA triangle is the side that lies directly between the two known angles. For example, if angle A is at vertex A and angle C is at vertex C, then side b (which connects vertex A to vertex C) is the included side. In standard triangle notation, side b is always opposite angle B, and it connects the vertices where angles A and C are located.

Banking Angle Calculator NZ

What is a banking angle?

It is the angle a road or track is tilted inward on a curve. Banking lets part of a vehicle's weight provide the inward force needed to corner, reducing or removing the reliance on tyre friction. The ideal angle depends on the design speed and the radius of the curve.

How is the ideal banking angle calculated?

It is the inverse tangent of the speed squared divided by the radius times gravity. At this angle the horizontal part of the road's normal force exactly supplies the centripetal force, so a vehicle at the design speed needs no friction to follow the curve.

Does banking work at all speeds?

The ideal angle is exact for one speed only. Below it, a vehicle tends to slide down the bank; above it, outward. In practice friction makes up the difference across the range of real traffic speeds, which is why banked curves are designed around a typical speed.

Basal Area Calculator

What is basal area in forestry?

Basal area is the cross-sectional area of a tree's stem measured at breast height (1.4 m above ground in New Zealand; some countries use 1.3 m and the United States uses 4.5 ft, about 1.37 m) and is expressed in square metres (m2) or square centimetres (cm2). Stand basal area is the total cross-sectional area of all trees in a defined area, usually expressed in square metres per hectare (m2/ha). It is a standard measure used by foresters to describe stand density, estimate timber volume, and guide thinning decisions. A stand basal area of 20 to 40 m2/ha is typical for productive commercial forests.

How do you measure DBH?

DBH stands for diameter at breast height. In New Zealand and most countries it is measured at 1.4 m above ground (some countries use 1.3 m). You can measure it directly with tree calipers or a diameter tape (D-tape). If you use a circumference tape, divide the circumference by pi (3.14159) to get the diameter. The measurement should be taken on the uphill side of the tree on sloping ground. On trees with buttresses, measure above the buttress.

What is a typical stand basal area for New Zealand plantation forests?

In New Zealand radiata pine plantations, stand basal area varies with age and management regime. Young stands may have a basal area of 5 to 10 m2/ha, while older stands approaching final harvest typically reach 50 to 70 m2/ha. Post-thinning residual basal area targets are commonly 15 to 25 m2/ha, depending on the intended end product. Native forests such as kauri or beech can have stand basal areas of 40 to 80 m2/ha. The basal area per hectare figure is also used to set stocking levels and assess competition for light and resources.

Base of a Triangle Calculator

How do you find the base of a triangle when you know the area and height?

Use the rearranged area formula: base = (2 x area) / height. For example, if a triangle has an area of 30 square centimetres and a perpendicular height of 6 cm, the base is (2 x 30) / 6 = 10 cm. This works for any triangle, whether it is scalene, isosceles, or equilateral, as long as the height is measured perpendicularly from the base to the opposite vertex.

Can you find the base of a triangle if you only know all three sides?

Yes. If you know all three side lengths (a, b, c) and you want to find the height to one of the sides (so that 'base' is that side), you can use Heron's formula to calculate the area first: s = (a + b + c) / 2, then Area = sqrt(s(s-a)(s-b)(s-c)). Once you have the area, use height = 2 x Area / base. If you want the actual length of a side as the base, the base is simply whichever side you designate as the base. The 'base' of a triangle is any side you choose; height is then the perpendicular distance to it.

How do you find the base of a right triangle?

For a right triangle, the two legs are perpendicular to each other. If you know one leg and the hypotenuse, use the Pythagorean theorem: base = sqrt(hypotenuse squared minus height squared). For example, if the hypotenuse is 13 and one leg (height) is 5, the base is sqrt(169 - 25) = sqrt(144) = 12. If you know both legs, either leg can act as the base.

Bladder Volume Calculator

What formula is used to calculate bladder volume?

Bladder volume is calculated using the ellipsoid formula: V = 0.523 × Length × Width × Height, where 0.523 is the mathematical constant π/6 (approximately). The three dimensions are typically obtained from an ultrasound scan or bladder scanner and entered in centimetres. The result is in cubic centimetres (cc), which is equal to millilitres (mL). This formula treats the bladder as a prolate ellipsoid, which closely approximates its shape when partially or moderately filled.

What is a normal post-void residual bladder volume?

A post-void residual (PVR) volume of less than 50 mL is generally considered normal and indicates effective bladder emptying. A PVR of 50 to 100 mL is in a borderline range and may warrant monitoring. A PVR greater than 200 mL is considered clinically significant and may indicate urinary retention or incomplete bladder emptying, which requires medical evaluation. Normal maximum bladder capacity is typically 300 to 500 mL in adults, though it varies with age and health status. These ranges are for reference only; a healthcare professional should interpret all results in clinical context.

How accurate is the ellipsoid formula for bladder volume?

The ellipsoid formula (V = 0.523 × L × W × H) is the most widely used method for estimating bladder volume from ultrasound measurements. Studies show it has a mean error of approximately 10 to 15 percent compared to catheterised volume, which is sufficient for most clinical decisions. Accuracy is highest when the bladder is moderately filled (100 to 400 mL) and tends to decrease at very low or very high volumes when the bladder shape deviates more from a true ellipsoid. Dedicated bladder scanners use the same mathematical principle but take multiple cross-sectional measurements to improve accuracy further.

Blast Radius Calculator

What does a blast radius calculator estimate?

It estimates the distance from an explosion at which the air blast reaches a chosen peak overpressure, the pressure spike above normal atmospheric pressure that causes injury and structural damage. You enter the charge weight as a TNT equivalent and a target overpressure, and it returns the radius at which that pressure level is reached, along with the scaled distance and the likely level of damage.

How is the blast distance calculated?

It uses the Kinney-Graham equation, a widely cited fit to measured blast data, which gives peak overpressure as a function of the scaled distance. Scaled distance is the true distance divided by the cube root of the charge weight, so distances scale with the cube root of the charge. The calculator solves the equation for the distance that produces your chosen overpressure.

How accurate is the estimate?

The scaling law is a good general model for a bare TNT charge in open air, but real events depend on the explosive type, confinement, height of burst, terrain and reflections. TNT equivalence itself is approximate. Treat the result as an order-of-magnitude estimate for education and planning, not a precise safety boundary.

Bolt Circle Calculator

What is a bolt circle diameter (PCD)?

The pitch circle diameter (PCD), also called bolt circle diameter (BCD), is the diameter of the imaginary circle that passes through the centre of each bolt hole in a bolt pattern. For example, a 4x100 wheel bolt pattern means 4 bolts equally spaced on a 100 mm diameter circle. To find the PCD on an existing part, measure the distance between the centres of two adjacent bolts and divide by sin(pi/N), where N is the number of bolts.

How do I calculate the XY coordinates of bolt holes?

For bolt hole i (numbered from 0), the coordinates relative to the circle centre are: x = (PCD/2) x cos(start_angle + i x 360/N) and y = (PCD/2) x sin(start_angle + i x 360/N), where N is the total number of bolts and angles are measured in degrees. This calculator does the trigonometry for you and displays the coordinates for each hole.

What is the chord length between adjacent bolts?

The chord length is the straight-line distance between the centres of two adjacent bolt holes. The formula is: chord = PCD x sin(pi/N), where N is the number of bolts and pi is approximately 3.14159. For example, 6 bolts on a 100 mm PCD gives a chord of 100 x sin(30 degrees) = 50.00 mm. This measurement is useful for verifying bolt positions with a calliper without needing to locate the circle centre.

Brewster's Angle Calculator

What is Brewster's angle?

Brewster's angle (also called the polarisation angle) is the specific angle of incidence at which light travelling between two media is reflected with no p-polarised (parallel-polarised) component. At this angle, the reflected ray is completely s-polarised (perpendicular polarised), and the p-polarised component passes entirely into the second medium. It is given by the formula theta_B = arctan(n2 / n1), where n1 is the refractive index of the first medium and n2 is the refractive index of the second medium.

What is the Brewster angle for air to glass?

For light passing from air (n1 = 1.000) into ordinary crown (window) glass (n2 = 1.500), Brewster's angle is arctan(1.500 / 1.000) = arctan(1.5) = approximately 56.31 degrees. This is the angle of incidence measured from the normal to the glass surface. At this angle, all reflected light is s-polarised. Different types of glass have slightly different refractive indices, so the exact Brewster angle varies a little.

Why does Brewster's angle matter in photography and optics?

Brewster's angle is the basis for polarising filters. When a camera's polarising filter is aligned to block s-polarised light, it cuts glare from water, glass, and other non-metallic surfaces most effectively when the camera is aimed at Brewster's angle to that surface (typically around 53 to 57 degrees for glass and water). Polarised sunglasses work on the same principle, cutting horizontal glare from roads and water by absorbing s-polarised light. In laser optics, Brewster windows allow a beam to pass through a glass window with zero reflection loss for p-polarised light.

Capsule Volume Calculator

What is the volume formula for a capsule?

A capsule consists of a cylinder with a hemisphere on each end. The total volume is pi times r squared times h plus (4/3) times pi times r cubed, which equals pi times r squared times (h + (4/3) r). Here r is the radius and h is the height of the cylindrical middle section only, not the total length.

What is the surface area of a capsule?

The surface area of a capsule is 2 pi r h plus 4 pi r squared. The first term is the lateral surface of the cylinder and the second term is the surface area of the complete sphere formed by joining the two hemispheres.

What is the total length of a capsule?

The total end-to-end length of a capsule is the cylinder height h plus the diameter of the two hemispherical caps: total length = h + 2r. The h in the formula refers only to the straight cylindrical section, not the full length.

Centre of a Circle Calculator

How do you find the centre of a circle from its general equation?

The general equation of a circle is x² + y² + Dx + Ey + F = 0. The centre is at (h, k) = (-D/2, -E/2), and the radius is r = sqrt((D/2)² + (E/2)² - F). For example, x² + y² - 6x + 4y - 12 = 0 gives D = -6, E = 4, F = -12, so the centre is (3, -2) and the radius is sqrt(9 + 4 + 12) = 5.

How do you find the centre of a circle from three points?

Three points on a circle uniquely determine the circle. Substituting each point (x, y) into x² + y² + Dx + Ey + F = 0 gives three linear equations in D, E, and F. Solving this system (using Cramer's rule or row reduction) yields D, E, and F, from which the centre (-D/2, -E/2) and radius sqrt((D/2)² + (E/2)² - F) can be found.

What is the standard form of a circle equation?

The standard (centre-radius) form is (x - h)² + (y - k)² = r², where (h, k) is the centre and r is the radius. The general form x² + y² + Dx + Ey + F = 0 can be converted to standard form by completing the square for both x and y. The two forms are related by h = -D/2, k = -E/2, and r² = h² + k² - F.

Central Angle Calculator

What is the formula for the central angle of a circle?

The central angle in radians equals the arc length divided by the radius: theta = arc length / radius. To convert to degrees, multiply by 180 and divide by pi: degrees = (arc length / radius) x (180 / pi). Alternatively, if you know the sector area and radius, the central angle in radians is: theta = (2 x sector area) / radius squared.

How do I find the central angle from the arc length and radius?

Divide the arc length by the radius to get the central angle in radians. For example, if the arc length is 7.854 and the radius is 5, the central angle is 7.854 / 5 = 1.5708 radians, which equals 90 degrees. The relationship comes from the definition of a radian: one radian is the angle subtended when the arc length equals the radius.

What is the difference between a central angle and an inscribed angle?

A central angle has its vertex at the centre of the circle, and its two sides (radii) extend to the circumference. An inscribed angle has its vertex on the circumference, with its two sides as chords of the circle. The inscribed angle theorem states that an inscribed angle is exactly half the central angle that subtends the same arc. So if a central angle is 90 degrees, an inscribed angle subtending the same arc is 45 degrees.

Centroid of a Triangle Calculator

What is the centroid of a triangle?

The centroid of a triangle is the point where its three medians intersect. A median is a line segment drawn from a vertex to the midpoint of the opposite side. The centroid is also the triangle's centre of mass: if you cut a triangle out of uniform material, it would balance perfectly on the centroid point. The centroid always lies inside the triangle, at two-thirds of the distance from each vertex to the midpoint of the opposite side.

How do you calculate the centroid of a triangle?

The centroid formula is straightforward: Cx = (x1 + x2 + x3) / 3 and Cy = (y1 + y2 + y3) / 3, where (x1, y1), (x2, y2) and (x3, y3) are the coordinates of the three vertices. You simply average all three x-coordinates to get the centroid x, and average all three y-coordinates to get the centroid y. For example, a triangle with vertices at (0, 0), (6, 0) and (3, 6) has centroid at ((0+6+3)/3, (0+0+6)/3) = (3, 2).

Where is the centroid relative to each vertex?

The centroid divides each median in a 2:1 ratio from vertex to midpoint. This means the centroid is located two-thirds of the way from any vertex to the midpoint of the opposite side, and one-third of the way from the midpoint of a side to the opposite vertex. This property holds for every median in the triangle, and the centroid is always inside the triangle regardless of its shape (acute, right, or obtuse).

Chi-Square Calculator

What is the Chi-Square Calculator?

Calculate chi-square statistics for observed and expected frequencies. Enter your category counts to get the chi-square value, degrees of freedom, and hypothesis test result at the 0.05 significance level.

Is the Chi-Square Calculator free to use?

Yes. The Chi-Square Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

How does the Chi-Square Calculator work?

Enter the number of categories and the observed frequency for each. The calculator works out the expected frequency, the chi-square statistic and the degrees of freedom, then compares the result against the critical value at the 0.05 significance level to tell you whether the null hypothesis can be rejected. With two categories it applies the Yates continuity correction.

Chi-Square Goodness of Fit Calculator NZ

What does the chi-square goodness-of-fit test do?

It tests whether observed frequencies differ from expected frequencies by more than chance. The statistic sums the squared differences between observed and expected, each divided by the expected. A large value suggests the data does not fit the expected distribution.

What are the degrees of freedom?

For a goodness-of-fit test, the degrees of freedom equal the number of categories minus one. If you estimated parameters from the data to form the expected values, you subtract one more for each. You use the degrees of freedom to find the critical value or p-value.

How do I judge significance?

Compare the chi-square statistic against the critical value from a chi-square table at your significance level and degrees of freedom, or find the p-value. If the statistic exceeds the critical value, the observed data differs significantly from the expected distribution.

Chi-Square Test of Independence

What is the expected count?

For each cell it is the row total times the column total divided by the grand total. It is the count you would expect if the two variables were completely independent. The test compares these with the counts you actually observed.

Why are the degrees of freedom one?

For a table with R rows and C columns the degrees of freedom are (R minus 1) times (C minus 1). A 2x2 table gives (2 minus 1) times (2 minus 1), which equals one. Once one cell is fixed the others are determined by the totals.

How do I decide if the result is significant?

Compare the chi-square statistic with a critical value from a chi-square distribution table at one degree of freedom. At the 5 percent level that value is about 3.84. A statistic larger than the critical value suggests the variables are not independent.

Circle Calculator

How do I find the area of a circle?

The area is pi times the radius squared. So a circle with a 5 unit radius has an area of pi times 25, about 78.5 square units. Enter the radius and the calculator gives the area, circumference, and diameter.

What is the circumference of a circle?

The circumference is the distance around the circle. It is pi times the diameter, or two times pi times the radius. For a 5 unit radius it is about 31.4 units.

What is the relationship between radius and diameter?

The diameter is twice the radius, the full distance across the circle through the centre. The radius is half the diameter. The calculator converts between them automatically.

Circle Formula Calculator

What is the formula for the area of a circle?

The area of a circle is A = pi times r squared, where r is the radius and pi is approximately 3.14159. For example, a circle with radius 5 cm has an area of pi x 25 = 78.54 cm squared. If you know the diameter instead of the radius, use A = pi x (d/2) squared, or equivalently A = pi x d squared divided by 4.

What is the formula for the circumference of a circle?

The circumference of a circle is C = 2 x pi x r, where r is the radius. This is equivalent to C = pi x d, where d is the diameter. For example, a circle with radius 5 cm has a circumference of 2 x pi x 5 = 31.42 cm. The circumference is also called the perimeter of the circle.

How do I find the radius if I know the circumference or area?

If you know the circumference C, the radius is r = C divided by (2 x pi). If you know the area A, the radius is r = the square root of (A divided by pi). For example, if the circumference is 31.416 cm, the radius is 31.416 divided by 6.2832 = 5 cm. If the area is 78.54 cm squared, the radius is the square root of (78.54 divided by 3.14159) = the square root of 25 = 5 cm.

Circle Length Calculator

What is the formula for the length (circumference) of a circle?

The circumference of a circle is found using C = 2πr, where r is the radius. If you know the diameter instead, the formula becomes C = πd, because the diameter is twice the radius (d = 2r). The constant π is approximately 3.14159265. For example, a circle with radius 5 cm has a circumference of 2 × π × 5 = 31.416 cm.

What is the difference between circumference and arc length?

The circumference is the total length around the complete circle (360°). Arc length is the length of any curved portion of the circle corresponding to a specific central angle. The arc length formula is L = (θ ÷ 360) × 2πr, where θ is the central angle in degrees. For a semicircle (θ = 180°), the arc length is exactly half the circumference.

How do I find the circumference if I only know the area?

If you know the area (A) of the circle, first find the radius using r = √(A ÷ π), then calculate the circumference using C = 2πr. For example, if the area is 78.54 cm², then r = √(78.54 ÷ π) = √25 = 5 cm, and the circumference is 2 × π × 5 = 31.42 cm.

Circle Measurements Calculator

How do I calculate the area of a circle from the diameter?

To find the area from a diameter, first halve the diameter to get the radius (r = d / 2), then apply the formula Area = pi x r squared. For example, a circle with diameter 10 has radius 5, so its area is pi x 5 x 5 = 78.54 square units. You can also use the shortcut formula Area = pi x (d / 2) squared directly.

What is the relationship between circumference and diameter?

The circumference of a circle equals pi multiplied by the diameter (C = pi x d). This relationship defines pi: it is the ratio of any circle's circumference to its diameter, which is always approximately 3.14159. Equivalently, circumference equals 2 x pi x radius (C = 2 x pi x r). To find diameter from circumference, divide by pi: d = C / pi.

Can I calculate all circle measurements from just one known value?

Yes. Because all four key circle measurements (radius, diameter, circumference, and area) are related through the constant pi, knowing any one value allows you to calculate the other three. The chain is: diameter = 2 x radius; circumference = 2 x pi x radius; area = pi x radius squared. Working backwards: if you know the area, radius = square root of (area / pi). If you know the circumference, radius = circumference / (2 x pi).

Circle Perimeter Calculator

What is the formula for the perimeter of a circle?

The perimeter of a circle is called its circumference. The formula is C = 2πr, where r is the radius, or equivalently C = πd where d is the diameter. Using the mathematical constant π (approximately 3.14159265), a circle with a radius of 5 cm has a circumference of 2 × 3.14159 × 5 = 31.416 cm.

What is the difference between circumference and perimeter of a circle?

There is no difference. The word 'perimeter' describes the total boundary length of any two-dimensional shape. For a circle specifically, this boundary is called the circumference. The two terms are used interchangeably when referring to a circle: circumference = perimeter = 2πr.

How do I find the perimeter of a circle if I only know the area?

If you know the area A of a circle, first find the radius using r = √(A / π), then calculate the circumference using C = 2πr. For example, if the area is 78.54 cm², the radius is √(78.54 / 3.14159) = √25 = 5 cm, and the circumference is 2 × 3.14159 × 5 = 31.42 cm.

Circle Segment Calculator

What is a circle segment?

A circle segment is the region between a chord and the arc it subtends. It is different from a sector, which is the pie-slice region between two radii. The segment area equals the sector area minus the triangle area. A semicircle is the special case where the central angle is 180 degrees, making the chord a diameter.

What is the sagitta of a segment?

The sagitta (from the Latin for arrow) is the perpendicular distance from the midpoint of the chord to the arc. It tells you how deep the segment is. For a central angle theta and radius r, the sagitta equals r minus r times cos(theta divided by 2), or r times (1 minus cos(theta/2)).

What is the formula for a circle segment area?

For a segment with radius r and central angle theta in radians: area = (r squared times (theta minus sin(theta))) divided by 2. In degrees, convert theta to radians first by multiplying by pi divided by 180. The formula subtracts the triangle area (r squared times sin(theta) divided by 2) from the sector area (r squared times theta divided by 2).

Circle Skirt Calculator

How do I calculate the inner radius for a circle skirt?

For a full circle skirt, the inner radius equals your waist measurement divided by (2 x pi), which is approximately 6.2832. For a half circle skirt, divide the waist by pi. For a quarter circle skirt, divide the waist by (pi / 2). Adding a seam allowance of 1 to 1.5 cm to the inner radius is recommended before cutting.

How much fabric do I need for a circle skirt?

The fabric you need depends on the circle type and your measurements. For a full circle skirt, you need a square of fabric at least as wide and tall as twice the outer radius (inner radius plus skirt length). The outer radius plus a seam allowance gives you the cutting radius. For example, a full circle skirt with a 70 cm waist and 60 cm length needs a fabric square of approximately 145 cm x 145 cm (before adding seam allowances). Fabric area without seam allowances is about 1.55 square metres.

What is the difference between a full, half, and quarter circle skirt?

A full circle skirt uses a complete 360-degree circle and creates the most flare and volume. A half circle skirt uses 180 degrees (two mirror-image pieces sewn together) and has moderate flare with less fabric. A quarter circle skirt uses 90 degrees (four pieces) and sits closer to the body with less flare. All three styles use the same inner radius formula, adjusted by the fraction of the circle being used.

Circumference and Area of a Circle Calculator

What is the formula for the circumference of a circle?

The circumference of a circle is calculated using C = 2πr, where r is the radius and π (pi) is approximately 3.14159. An equivalent form using diameter is C = πd, where d = 2r. For example, a circle with a radius of 5 cm has a circumference of 2 × π × 5 = 31.416 cm.

What is the formula for the area of a circle?

The area of a circle is calculated using A = πr², where r is the radius and π is approximately 3.14159. For example, a circle with a radius of 5 cm has an area of π × 5² = π × 25 = 78.540 cm². If you know the diameter instead of the radius, use A = π(d/2)² or equivalently A = πd²/4.

How do I find the radius from the circumference?

Rearrange C = 2πr to give r = C / (2π). For example, if the circumference is 31.416 cm, the radius is 31.416 / (2 × 3.14159) = 5 cm. Once you have the radius, you can calculate the area as A = πr². This calculator accepts circumference as an input and computes all other properties automatically.

Circumference Calculator

What is the formula for the circumference of a circle?

The circumference of a circle is calculated using C = 2πr (where r is the radius) or equivalently C = πd (where d is the diameter). Pi (π) is the constant approximately equal to 3.14159. For example, a circle with a radius of 5 cm has a circumference of 2 × 3.14159 × 5 = 31.416 cm.

How do I find the radius from the circumference?

Rearrange C = 2πr to get r = C / (2π). For example, if the circumference is 31.416 cm, the radius is 31.416 / (2 × 3.14159) = 5 cm. The diameter is simply twice the radius: d = C / π.

What is the difference between circumference and perimeter?

Circumference is the specific term for the perimeter of a circle. Perimeter is the general term for the total boundary length of any closed shape. For a circle, circumference and perimeter refer to the same measurement: the total distance around the outside of the circle.

Circumference of a Cylinder Calculator

What is the circumference of a cylinder?

The circumference of a cylinder is the perimeter of its circular base. It is calculated using the formula C = 2 x pi x r, where r is the radius of the base circle. Equivalently, C = pi x d where d is the diameter. For a cylinder with radius 5 cm, the circumference is 2 x 3.14159 x 5 = 31.416 cm.

Why does the height of the cylinder not affect the circumference?

The circumference of a cylinder refers to the circular cross-section, not the overall shape. Since a circle's perimeter depends only on its radius (C = 2 x pi x r), the height of the cylinder has no effect on the base circumference. Height does matter when calculating lateral surface area (2 x pi x r x h) or volume (pi x r squared x h), but not circumference.

How do I find the radius of a cylinder if I know the circumference?

Rearrange the circumference formula C = 2 x pi x r to solve for r: r = C divided by (2 x pi). For example, if the circumference is 62.832 cm, then r = 62.832 / (2 x 3.14159) = 10 cm. You can use this calculator in reverse: enter the circumference in the radius field after dividing it by 2 x pi first, or simply use the rearranged formula directly.

Circumference to Diameter Calculator

How do I find the diameter from the circumference?

Divide the circumference by pi (approximately 3.14159). The formula is d = C / pi. For example, a circle with a circumference of 31.416 cm has a diameter of 31.416 / 3.14159 = 10 cm. This works in any unit: millimetres, centimetres, metres, inches, or feet, as long as the circumference and diameter are in the same unit.

What is the relationship between circumference, diameter, and radius?

The circumference C of a circle equals pi times the diameter d: C = pi x d. The diameter is twice the radius (d = 2r), so C = 2 x pi x r as well. To reverse this: d = C / pi and r = C / (2 x pi). All three measurements are proportional to each other through pi.

Can I also get the area from the circumference?

Yes. First find the radius r = C / (2 x pi), then calculate the area as A = pi x r squared. This calculator does both steps automatically. For example, a circumference of 62.832 cm gives a radius of 10 cm and an area of 314.159 square centimetres.

Circumscribed Circle Calculator

What is the circumscribed circle of a triangle?

The circumscribed circle (also called the circumcircle) of a triangle is the unique circle that passes through all three vertices of the triangle. Its centre is called the circumcentre and its radius is called the circumradius (R). Every triangle has exactly one circumscribed circle. The circumradius is given by R = abc / (4A), where a, b, c are the side lengths and A is the area of the triangle.

How do you calculate the circumradius of a triangle?

The circumradius R is calculated using the formula R = (a x b x c) / (4 x A), where a, b and c are the three side lengths and A is the area of the triangle. The area is found using Heron's formula: A = sqrt(s x (s - a) x (s - b) x (s - c)), where s is the semi-perimeter s = (a + b + c) / 2. For a right-angled triangle the circumradius equals half the hypotenuse, so R = hypotenuse / 2.

What is the circumradius of a right-angled triangle?

For a right-angled triangle, the circumscribed circle has a diameter equal to the hypotenuse. This means the circumradius R = hypotenuse / 2. For example, a 3-4-5 right triangle has a hypotenuse of 5, so R = 5 / 2 = 2.5. The circumcentre is located at the midpoint of the hypotenuse.

Clock Angle Calculator

What is the angle between clock hands at 3:00?

At exactly 3:00, the hour hand points to 90 degrees (the 3 o'clock position) and the minute hand points to 0 degrees (the 12 o'clock position). The angle between them is exactly 90 degrees. This is a classic example often used in maths problems because the hands form a perfect right angle.

How do you calculate the angle between clock hands?

Use these two formulas. Hour hand angle (from 12 o'clock) = (H mod 12) x 30 + M x 0.5, where H is the hour and M is the minutes. This is because each hour mark is 30 degrees apart (360 / 12) and the hour hand also advances 0.5 degrees per minute. Minute hand angle = M x 6, because the minute hand moves 6 degrees per minute (360 / 60). The angle between the hands is the absolute difference between these two values. If the result is greater than 180 degrees, subtract it from 360 to get the smaller angle.

At what times are the clock hands exactly together (0 degrees)?

The hands coincide (0 degree angle) 11 times in every 12 hours, not 12. Starting from 12:00, the hands meet again at approximately 1:05:27, 2:10:54, 3:16:21, 4:21:49, 5:27:16, 6:32:43, 7:38:10, 8:43:38, 9:49:05, 10:54:32, and then back at 12:00. The minute hand gains on the hour hand at a rate of 5.5 degrees per minute, so each coincidence occurs every 720/11 minutes (approximately 65 minutes and 27 seconds).

Completing the Square Calculator

What does completing the square do?

Completing the square rewrites a quadratic expression in standard form (ax squared plus bx plus c) into vertex form a(x minus h) squared plus k. This reveals the vertex of the parabola at (h, k) and makes it straightforward to read off the axis of symmetry, maximum or minimum value, and roots.

How do you complete the square step by step?

For x squared plus bx plus c: take half of b (call it p), then add and subtract p squared. The expression becomes (x plus p) squared minus p squared plus c. For example, x squared minus 6x plus 5 becomes (x minus 3) squared minus 9 plus 5, which is (x minus 3) squared minus 4.

Can you complete the square when a is not 1?

Yes. Factor out a from the first two terms first: ax squared plus bx plus c equals a(x squared plus (b/a)x) plus c. Then complete the square inside the bracket and adjust the constant term accordingly. The vertex h is always minus b divided by 2a, and k is c minus b squared divided by 4a.

Concrete Cylinder Calculator

How do I calculate the volume of concrete for a cylinder?

Use the cylinder volume formula: Volume = pi x radius squared x height (V = pi r2 h). For a post hole with a diameter of 300 mm (radius = 0.15 m) and a depth of 600 mm (0.6 m), the volume is 3.14159 x 0.15 x 0.15 x 0.6 = approximately 0.042 cubic metres (42 litres). Multiply that by the concrete density (typically 2,400 kg per cubic metre) to get the weight, then divide by your bag size to find how many bags you need.

How many 20 kg bags of premix concrete fill a post hole?

A single 20 kg bag of standard premix concrete yields approximately 10 litres (0.010 cubic metres) of mixed concrete. For a 300 mm diameter hole that is 600 mm deep, the volume is about 42.4 litres, requiring roughly 5 bags (allowing for the volume displaced by the post itself). Always buy one or two extra bags to account for wastage and spillage. For sonotubes or larger diameter fills, the number of bags rises quickly, so consider ordering ready-mix concrete for holes larger than about 0.1 cubic metres.

What is a sonotube and how much concrete does it hold?

A sonotube (or concrete form tube) is a cylindrical cardboard tube used as a mould for pouring concrete columns, piers, and footings. Common sizes in New Zealand range from 100 mm to 600 mm diameter. A 200 mm sonotube that is 1 metre tall holds approximately 31.4 litres of concrete (0.031 m3), which weighs about 75 kg and requires 4 bags of 20 kg premix. A 300 mm sonotube at the same height holds about 70.7 litres, requiring 8 bags. Use this calculator to find the exact volume for any diameter and height.

Concrete Volume Calculator

How many bags are in a cubic metre?

It depends on the bag yield, which is printed on the bag. A 20 kilogram premix bag typically yields about 0.01 to 0.012 cubic metres, so a cubic metre takes roughly 83 to 100 bags. Always check your specific product.

Should I order extra concrete?

Yes. It is wise to add about 5 to 10 percent on top of the calculated volume to cover spillage, uneven ground and slightly over-dug formwork. Running short during a pour is far more costly than a small amount of waste.

When should I use ready-mix instead of bags?

For anything more than a few bags, ready-mix delivered by the cubic metre is usually cheaper, faster and gives a more consistent result. Bags suit small jobs like setting a few posts or patching. Large slabs are best poured from a truck.

Cone Surface Area Calculator

What is the formula for the lateral surface area of a cone?

Lateral surface area = pi x radius x slant height. The slant height is the distance from the apex to the edge of the base, equal to the square root of (radius squared + height squared). For r=4 and h=3: slant = 5, lateral SA = pi x 4 x 5 = 62.83 square units.

How do you find the total surface area of a cone?

Add the lateral surface area to the circular base: total SA = pi x r x slant + pi x r squared = pi x r x (slant + r). For r=4 and slant=5: total SA = pi x 4 x 9 = 113.10 square units.

What is the slant height of a cone?

The slant height is the straight-line distance from the apex of the cone down the outside to any point on the base edge. It is longer than the vertical height and is found using Pythagoras: slant = sqrt(r squared + h squared). For r=4 and h=3: slant = sqrt(16+9) = sqrt(25) = 5.

Cone Volume Calculator

What is the volume of a cone?

The volume of a cone is one third times pi times the radius squared times the height. It is exactly a third of a cylinder with the same base and height. Enter the radius and height and the calculator works it out.

What is the slant height?

The slant height is the distance from the edge of the base to the tip along the surface. It is the square root of the radius squared plus the height squared, by Pythagoras, and is used for the surface area.

How do I find the surface area of a cone?

The total surface area is pi times the radius times the slant height, plus pi times the radius squared for the base. The calculator includes the base in the figure it shows.

Cosine Calculator

How do I find the cosine of an angle?

Enter the angle and pick degrees or radians. The calculator converts to radians and evaluates the cosine.

Degrees or radians?

Either. Choose the unit in the dropdown. The angle is shown in both units beside the result so you can check.

Why is the result the same every 360 degrees?

The trig functions are periodic. Sine and cosine repeat every 360 degrees (2 pi radians), so equivalent angles give the same value.

Cosine Triangle Calculator (Law of Cosines)

What is the Law of Cosines?

The Law of Cosines relates the lengths of the three sides of a triangle to the cosine of one of its angles. The formula is c² = a² + b² - 2ab·cos(C), where a, b, and c are the side lengths and C is the angle opposite side c. It generalises Pythagoras's theorem: when C is 90 degrees, cos(C) equals zero and the formula reduces to c² = a² + b². You can rearrange the formula to find an unknown angle: cos(C) = (a² + b² - c²) / (2ab).

When should I use the Law of Cosines instead of the Law of Sines?

Use the Law of Cosines when you know two sides and the included angle (SAS configuration), or when you know all three sides (SSS configuration). The Law of Sines is more convenient when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA, though this case can have two solutions). The Law of Cosines avoids the ambiguous case that arises with SSA using the Law of Sines.

How do I find all angles when I know all three sides?

When all three sides are known (SSS), rearrange the Law of Cosines to solve for each angle. For angle A: cos(A) = (b² + c² - a²) / (2bc). For angle B: cos(B) = (a² + c² - b²) / (2ac). For angle C: cos(C) = (a² + b² - c²) / (2ab). Apply the inverse cosine (arccos) to each result to get the angle in degrees. The three angles must sum to exactly 180 degrees. If the cosine value falls outside the range -1 to 1, the three side lengths cannot form a valid triangle.

Coterminal Angle Calculator

What are coterminal angles?

Coterminal angles are angles that share the same terminal side when drawn in standard position on a coordinate plane. You get them by adding or subtracting any multiple of 360 degrees from the original angle. For example, 30 degrees, 390 degrees and minus 330 degrees are all coterminal because they all end at the same position on the unit circle.

How do you find the principal coterminal angle?

The principal coterminal angle is the equivalent angle between 0 degrees and 360 degrees (exclusive). To find it, divide the angle by 360, take the remainder (modulo 360) and adjust for negatives. For example, 390 mod 360 = 30 degrees. For minus 30 degrees, add 360 to get 330 degrees.

Can coterminal angles be negative?

Yes. A negative coterminal angle is found by subtracting 360 degrees (or a multiple of 360) from the principal angle. For 30 degrees, subtracting 360 gives minus 330 degrees, which is a valid negative coterminal. Both angles end at the same position on the unit circle.

Critical Angle Calculator NZ

What is the critical angle?

It is the angle of incidence, measured from the perpendicular, beyond which light travelling from a denser to a less dense medium cannot escape and is totally internally reflected. At exactly the critical angle, the refracted ray would skim along the boundary.

What is total internal reflection?

When light strikes a boundary toward a less dense medium at more than the critical angle, none of it refracts out; it is all reflected back into the denser medium. This traps light, which is how optical fibres guide signals and why cut diamonds sparkle.

When is there no critical angle?

Only light going from a denser to a less dense medium, where the first index is larger, has a critical angle. Going the other way, into a denser medium, light can always refract out, so total internal reflection and a critical angle do not exist for that direction.

Cross-Sectional Area Calculator

What is cross-sectional area?

Cross-sectional area is the area of a two-dimensional slice taken perpendicular to an object's length or axis. For example, if you cut a steel rod straight across, the exposed circle is its cross-section. The cross-sectional area determines how much material is present at that slice, which is why it is used in structural engineering, fluid mechanics, and physics to calculate stress (force divided by area), flow rate, and electrical resistance.

How do you calculate the cross-sectional area of a circle?

The cross-sectional area of a circle is A = pi x r^2, where r is the radius. If you know the diameter d instead, substitute r = d/2 to get A = pi x (d/2)^2. For example, a circular pipe with an inner radius of 5 cm has a cross-sectional area of pi x 25 = 78.54 cm^2. This formula applies to any solid circular cross-section such as round bars, pipes (using the inner radius for flow area), or columns.

Why does cross-sectional area matter in engineering?

In structural engineering, stress is calculated as force divided by cross-sectional area (sigma = F / A). A larger cross-sectional area distributes a given load over more material, reducing stress. This is why thick columns and wide beams carry more load than thin ones of the same material. In fluid mechanics, flow rate equals velocity multiplied by cross-sectional area (Q = v x A), so the area of a pipe determines how much fluid passes through per second at a given velocity. In electrical engineering, a thicker wire has a larger cross-sectional area, which lowers its resistance.

Cube Calculator

What is the formula for the volume of a cube?

The volume of a cube is side length cubed: V = s^3. If the side is 5, the volume is 5 x 5 x 5 = 125 cubic units.

How do you find the surface area of a cube?

A cube has six identical square faces. The total surface area is 6 times the area of one face: SA = 6 x s^2. For a cube with side 5, SA = 6 x 25 = 150 square units.

What is the space diagonal of a cube?

The space diagonal runs from one corner of the cube to the opposite corner, passing through the interior. Its length is s times the square root of 3. For a cube with side 5, the space diagonal is 5 x 1.732 = 8.66 units.

Cube Density Calculator

How do you calculate the density of a cube?

Density equals mass divided by volume (density = mass / volume). For a cube, the volume is the side length cubed (volume = s x s x s = s3). So the full formula is: density = mass / s3. For example, a cube with a side of 5 cm and a mass of 337.5 g has a volume of 125 cm3 and a density of 337.5 / 125 = 2.7 g/cm3, which is close to the density of aluminium.

What units should I use for cube density?

The most common unit for density is grams per cubic centimetre (g/cm3), which is used for small objects. For larger engineering applications, kilograms per cubic metre (kg/m3) is standard. The two are related: 1 g/cm3 = 1000 kg/m3. Water has a density of 1 g/cm3 (1000 kg/m3), so materials with a density above 1 g/cm3 sink in water and those below float. Make sure your mass and side length use consistent units before dividing.

How do I find the side length of a cube if I know its density and mass?

Rearrange the density formula to find volume first: volume = mass / density. Since volume = s3 for a cube, the side length is the cube root of the volume: s = (mass / density)^(1/3). For example, if an aluminium cube (density 2.7 g/cm3) has a mass of 337.5 g, then volume = 337.5 / 2.7 = 125 cm3, and side = 125^(1/3) = 5 cm.

Cube Root Calculator

What is a cube root?

The cube root of a number x is the value y such that y multiplied by itself three times equals x. It is written as the cube root symbol (a radical with a small 3) over x, or equivalently as x raised to the power of one third. For example, the cube root of 27 is 3 because 3 times 3 times 3 equals 27.

Can a negative number have a cube root?

Yes. Unlike square roots, cube roots of negative numbers are real. The cube root of negative 27 is negative 3, because negative 3 multiplied by itself three times gives negative 27. This is because multiplying three negative numbers always yields a negative result.

How do you simplify a cube root?

To simplify a cube root, factorise the radicand and pull out any perfect cube factors. For example, the cube root of 54 can be written as the cube root of 27 times 2, which simplifies to 3 times the cube root of 2. If no perfect cube factor exists other than 1, the surd is already in simplest form.

Cube Surface Area and Volume Calculator

What is the formula for the surface area of a cube?

The surface area of a cube is SA = 6s², where s is the side length. A cube has six identical square faces, each with area s². Multiplying by six gives the total surface area. For example, a cube with side length 5 cm has a surface area of 6 × 25 = 150 cm².

What is the formula for the volume of a cube?

The volume of a cube is V = s³, where s is the side length. Because a cube has equal length, width, and height, the volume is simply the side length multiplied by itself three times. For example, a cube with side length 5 cm has a volume of 5 × 5 × 5 = 125 cm³.

How do you find the diagonal of a cube?

A cube has two types of diagonal. The face diagonal runs across one square face and equals s√2 (approximately 1.414 × s). The space diagonal runs from one corner of the cube to the opposite corner through the interior and equals s√3 (approximately 1.732 × s). For a cube with side 5 cm, the face diagonal is 7.071 cm and the space diagonal is 8.66 cm.

Cuboid Calculator

What is the formula for the volume of a cuboid?

Volume = Length x Width x Height. For a box measuring 4 x 3 x 5 units, the volume is 4 x 3 x 5 = 60 cubic units.

How do you calculate the surface area of a rectangular box?

A cuboid has three pairs of opposite rectangular faces. Total surface area = 2 x (LW + LH + WH). For L=4, W=3, H=5: SA = 2 x (12 + 20 + 15) = 2 x 47 = 94 square units.

What is the space diagonal of a cuboid?

The space diagonal is the straight-line distance from one corner of the box to the opposite corner through the interior. It equals the square root of (L squared + W squared + H squared). For L=4, W=3, H=5: diagonal = sqrt(16 + 9 + 25) = sqrt(50) = 7.07 units.

Cuboid Surface Area Calculator

What is the formula for the surface area of a cuboid?

The total surface area of a cuboid is SA = 2(lw + lh + wh), where l is length, w is width, and h is height. A cuboid has six rectangular faces: two faces of lw (top and bottom), two of lh (front and back), and two of wh (left and right). Summing all six faces gives the total surface area.

What is the difference between a cuboid and a cube?

A cube is a special case of a cuboid where all three dimensions (length, width, and height) are equal. The surface area of a cube simplifies to 6s², where s is the side length. For a general cuboid, the three dimensions can be different, so the full formula SA = 2(lw + lh + wh) is needed.

What units does this cuboid surface area calculator use?

The calculator works with any consistent unit. If you enter dimensions in centimetres, the surface area is in square centimetres (cm²). If you use metres, the result is in square metres (m²). Simply choose a unit label from the dropdown to display the correct unit alongside your result.

Cuboid Volume Calculator

What is the formula for the volume of a cuboid?

The volume of a cuboid is calculated by multiplying its three dimensions together: Volume = length x width x height (V = l x w x h). All three measurements must be in the same unit before multiplying. The result is expressed in cubic units, for example cubic metres (m³), cubic centimetres (cm³), or cubic feet (ft³). A cuboid is also known as a rectangular prism or rectangular box.

How do I convert the volume from cubic metres to litres?

One cubic metre equals exactly 1,000 litres. To convert, multiply your volume in m³ by 1,000. For example, a cuboid with volume 0.5 m³ holds 500 litres. Similarly, one cubic centimetre equals one millilitre, so a volume in cm³ gives you the same number of millilitres directly. This relationship is useful for calculating tank or container capacity.

What is the difference between a cube and a cuboid?

A cube is a special type of cuboid where all three dimensions (length, width and height) are equal. Every cube is a cuboid, but most cuboids are not cubes. Both have six rectangular faces, twelve edges and eight vertices. For a cube with side s, the volume formula simplifies to V = s³. For a general cuboid where the three dimensions differ, you must multiply all three separately: V = l x w x h.

Cylinder Surface Area Calculator

What is the formula for the lateral surface area of a cylinder?

The lateral (curved) surface area of a cylinder is 2 x pi x radius x height. If you unroll the curved surface it becomes a rectangle with width equal to the circumference and height equal to the cylinder height. For r=5 and h=10: lateral SA = 2 x pi x 5 x 10 = 314.16 square units.

How do you find the total surface area of a cylinder?

Add the lateral surface area to the two circular ends: total SA = 2 x pi x r x h + 2 x pi x r squared = 2 x pi x r x (r + h). For r=5 and h=10: total SA = 2 x pi x 5 x 15 = 471.24 square units.

What is the difference between lateral and total surface area?

The lateral surface area covers only the curved side of the cylinder, without the top and bottom circles. The total surface area includes both circular ends as well. Use lateral area when calculating material for a tube or pipe with open ends, and total area when the cylinder is a closed can or tank.

Cylinder Volume Calculator

What is the formula for the volume of a cylinder?

The volume of a cylinder is pi times the radius squared, times the height. In symbols, V = pi r squared h. If you know the diameter, the radius is half the diameter.

How do I convert cubic metres to litres?

One cubic metre is 1,000 litres. So multiply cubic metres by 1,000 to get litres. A cylinder of 0.5 cubic metres holds 500 litres.

How do I find the volume of a tank or pipe?

A tank or pipe is a cylinder, so use the same formula. Measure the internal radius and the length or height, then multiply pi by the radius squared by the length. Use the internal measurements for capacity.

Cylinder Volume in Litres Calculator

How do I work out the litres in a cylinder?

Find the volume in cubic centimetres using V = pi times radius squared times height, then divide by 1,000 to get litres (since 1 litre equals 1,000 cubic centimetres, and 1 cubic metre equals 1,000 litres). If you measured in millimetres, convert to centimetres first by dividing by 10, or let a calculator handle the unit conversion for you.

Can I calculate cylinder volume from circumference instead of radius?

Yes. If you have wrapped a tape measure around a cylinder and only know the circumference, first find the radius using r = circumference / (2 x pi). Then use the usual formula, V = pi r squared h, to get the volume. This is useful for round tanks, pots, or pipes where the diameter is hard to measure directly.

How many litres are in a standard 200 litre drum?

A standard 44 gallon (200 litre) drum is close to 200 litres by design, with an internal diameter of around 58 cm and a height of around 88 cm. Using V = pi r squared h with r = 29 cm and h = 88 cm gives approximately 232,500 cubic centimetres, or about 232 litres, which is slightly more than the nominal 200 litre rating because real drums allow headspace and the true dimensions vary by manufacturer.

Density Mass Volume Calculator

What is the formula linking density, mass and volume?

Density equals mass divided by volume: density = mass / volume. Rearranged, mass = density x volume, and volume = mass / density. Density is usually expressed in kilograms per cubic metre (kg/m3) or grams per cubic centimetre (g/cm3), where 1 g/cm3 is equal to 1000 kg/m3.

How do I find the volume of an irregular object to calculate its density?

Weigh the object to get its mass, then measure its volume by water displacement. Submerge the object fully in a measuring cylinder of water and record how much the water level rises; that rise in millilitres equals the object's volume in cubic centimetres. Divide the mass by this volume to get density.

Why do objects with the same volume have different masses?

Mass depends on both volume and density, and different materials pack different amounts of matter into the same space. A cubic centimetre of lead has a much higher density (about 11.34 g/cm3) than a cubic centimetre of aluminium (about 2.70 g/cm3), so even though the volumes are identical, the lead is over four times heavier.

Density of a Cylinder Calculator

What is the formula for the density of a cylinder?

Density equals mass divided by volume (rho = m / V). For a cylinder, the volume is V = pi x r^2 x h, where r is the radius and h is the height. So density = m / (pi x r^2 x h). Make sure your mass and volume units are consistent (for example, grams and cubic centimetres, or kilograms and cubic metres) before dividing.

How do I find the volume of a cylinder from its diameter?

If you know the diameter instead of the radius, divide the diameter by 2 to get the radius, then use V = pi x r^2 x h. For example, a cylinder with a 10 cm diameter and 20 cm height has a radius of 5 cm, giving a volume of pi x 5^2 x 20 = 1,570.8 cubic centimetres.

What units should I use for density?

Density is most commonly expressed in kilograms per cubic metre (kg/m3) for engineering and scientific use, or grams per cubic centimetre (g/cm3) for smaller lab samples. Water has a density of 1,000 kg/m3 (1 g/cm3), which is a useful benchmark: materials denser than water sink in it, and materials less dense float.

Diagonal of a Rectangle Calculator

What is the formula for the diagonal of a rectangle?

The diagonal of a rectangle is found using the Pythagorean theorem: diagonal = square root of (length squared plus width squared), written as d = sqrt(l^2 + w^2). This works because a rectangle's diagonal splits it into two right-angled triangles, with the diagonal as the hypotenuse and the length and width as the two shorter sides.

Do both diagonals of a rectangle have the same length?

Yes. A rectangle has two diagonals and they are always equal in length, because opposite sides of a rectangle are equal and each diagonal forms a right-angled triangle with the same pair of side lengths. This is one of the defining properties of a rectangle, and it is also true for squares since a square is a special rectangle.

How do I find the length or width if I know the diagonal?

Rearrange the Pythagorean theorem. If you know the diagonal (d) and one side, the other side is found using: missing side = square root of (d^2 minus known side^2). For example, if the diagonal is 5 m and the length is 4 m, the width is the square root of (25 minus 16), which is 3 m.

Diagonal of a Square Calculator

What is the formula for the diagonal of a square?

The diagonal of a square is d = s times the square root of 2, where s is the side length. This comes from Pythagoras's theorem, since a square's diagonal splits it into two right-angled triangles with the two equal sides as the legs and the diagonal as the hypotenuse. For example, a square with a 5 metre side has a diagonal of 5 times 1.41421, which is about 7.07 metres.

How do you find the side length from the diagonal?

Rearrange the diagonal formula to solve for the side: s = d divided by the square root of 2, which is the same as s = d times 0.70711. For example, a square with a diagonal of 10 metres has a side length of 10 divided by 1.41421, which is about 7.07 metres.

Is the diagonal of a square always longer than its side?

Yes. Because the square root of 2 is approximately 1.41421, the diagonal is always about 41.4 percent longer than the side length. This ratio holds for every square, regardless of size, since all squares are similar shapes with the same proportions.

Diameter of a Circle Calculator

How do you find the diameter of a circle from the radius?

The diameter is always twice the radius: d = 2r. The radius is the distance from the centre of the circle to its edge, and the diameter is the full distance across the circle through the centre. For example, a circle with a radius of 5 cm has a diameter of 10 cm.

How do you find the diameter of a circle from the area?

Rearrange the area formula A = pi r^2 to solve for the radius, then double it. First divide the area by pi and take the square root to get the radius: r = sqrt(A / pi). Then multiply by 2 to get the diameter: d = 2 x sqrt(A / pi). For example, a circle with an area of 78.54 square cm has a diameter of about 10 cm.

How do you find the diameter of a circle from the circumference?

Divide the circumference by pi: d = C / pi. Circumference and diameter are directly proportional, and pi (approximately 3.14159) is the constant ratio between them. For example, a circle with a circumference of 31.42 cm has a diameter of 10 cm.

Diameter to Radius Calculator

How do you convert diameter to radius?

Divide the diameter by 2. The radius is always exactly half the diameter, because the diameter is a straight line passing through the centre of a circle from one edge to the other, while the radius runs from the centre to the edge. For example, a circle with a diameter of 20 cm has a radius of 10 cm.

How do you convert radius to diameter?

Multiply the radius by 2. For example, a circle with a radius of 7 m has a diameter of 14 m. This is the reverse of the diameter to radius conversion, since diameter = 2 x radius always holds true for any circle.

Can I find the circumference and area from the radius or diameter?

Yes. Once you have the radius, the circumference equals 2 x pi x radius, and the area equals pi x radius squared. This calculator works out both automatically once you enter either the diameter or the radius, using pi to full JavaScript double precision (approximately 3.14159265358979).

Dimensions of a Rectangle Calculator

How do you find the length and width of a rectangle from area and perimeter?

If you know the area (A) and perimeter (P) of a rectangle, first find the half-perimeter: s = P / 2. This equals length + width. The two dimensions are then the two roots of the quadratic equation x^2 - s*x + A = 0, solved using x = (s +/- sqrt(s^2 - 4A)) / 2. The larger root is the length and the smaller root is the width. For example, with area 20 and perimeter 18, s = 9, and solving the quadratic gives dimensions of 5 and 4.

Can you find both dimensions from area and one side alone?

Yes. If you know the area (A) and one side (say the length, L), the other side (width) is simply W = A / L. This is the direct rearrangement of the area formula A = L x W. No quadratic equation is needed in this case because you already have one of the two unknowns.

What if the area and perimeter given do not produce a real rectangle?

A valid rectangle only exists if the discriminant of the quadratic (s^2 - 4A, where s is half the perimeter) is zero or positive. If the discriminant is negative, no real rectangle has that combination of area and perimeter, because the perimeter is too small to enclose that much area. If the discriminant is exactly zero, the rectangle is actually a square, since both roots are equal.

Double Angle Calculator

What is the double angle formula for sine?

sin(2x) = 2 sin x cos x. It follows from the sum formula with both angles equal to x.

Why does cos(2x) have three forms?

cos(2x) = cos squared x minus sin squared x, which rearranges using the Pythagorean identity into 2 cos squared x minus 1, or 1 minus 2 sin squared x.

When is tan(2x) undefined?

When tan squared x equals 1, that is at x = 45, 135 degrees and so on, where cos(2x) is zero.

Double Angle Formula Calculator

What are the double angle formulas?

The three double angle formulas are: sin(2x) = 2 sin x cos x, cos(2x) = cos squared x minus sin squared x (which also equals 2 cos squared x minus 1, or 1 minus 2 sin squared x), and tan(2x) = 2 tan x divided by (1 minus tan squared x). Each one comes from the angle sum formula with both angles set equal to x.

Why does cos(2x) have three different versions?

The base form cos(2x) = cos squared x minus sin squared x can be rewritten using the Pythagorean identity sin squared x plus cos squared x = 1. Substituting sin squared x = 1 minus cos squared x gives cos(2x) = 2 cos squared x minus 1. Substituting cos squared x = 1 minus sin squared x gives cos(2x) = 1 minus 2 sin squared x. All three forms always give the same numeric answer, so you choose whichever is more convenient for the problem you are solving.

When is the tan(2x) formula undefined?

tan(2x) = 2 tan x / (1 minus tan squared x) is undefined whenever the denominator is zero, which happens when tan x = 1 or tan x = -1, that is at x = 45 degrees, 135 degrees, 225 degrees, 315 degrees and every 90 degree interval from there. It is also undefined whenever x itself makes tan x undefined, at x = 90 degrees plus any multiple of 180 degrees.

Double Angle Identities Calculator

What are the double angle identities?

The double angle identities express sin(2x), cos(2x) and tan(2x) in terms of sin x, cos x and tan x. sin(2x) = 2 sin x cos x. cos(2x) = cos squared x minus sin squared x, which is also equal to 2 cos squared x minus 1, and to 1 minus 2 sin squared x. tan(2x) = 2 tan x divided by (1 minus tan squared x). They all follow from the angle sum identities sin(A + B) and cos(A + B) with A and B both set to x.

How do you check a double angle identity is correct?

Pick a test angle, work out sin x, cos x and tan x, then calculate 2x directly (for example using a calculator's sin, cos and tan buttons on the doubled angle) and compare it with the value produced by the identity. If both methods give the same number, the identity holds for that angle. Because these identities are true for every real value of x, testing several different angles (including negative angles and angles past 90 degrees) is a reliable way to confirm you have applied the identity correctly.

Why do the three cos(2x) forms always agree?

All three forms come from the same identity, cos(2x) = cos squared x minus sin squared x, rewritten using the Pythagorean identity sin squared x plus cos squared x = 1. Because sin squared x and cos squared x are never independent (one is always 1 minus the other), substituting one for the other cannot change the final numeric value. If a calculation shows the three forms disagreeing, it means there is a rounding or input error rather than a flaw in the identity itself.

Ellipse Calculator

What is the formula for the area of an ellipse?

The area of an ellipse is pi times a times b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). For an ellipse with a = 5 and b = 3, area = pi times 5 times 3 = approximately 47.12 square units.

How do you calculate the circumference of an ellipse?

Unlike a circle, there is no simple exact formula for the circumference of an ellipse. The best practical approximation is Ramanujan's formula: pi times (3(a+b) minus the square root of ((3a+b)(a+3b))). This is accurate to within about 0.0001% for most practical ellipses and is significantly better than the simple pi times (a+b) estimate.

What is eccentricity of an ellipse?

Eccentricity (e) measures how far an ellipse deviates from a circle. It is calculated as the square root of (1 minus (b squared divided by a squared)), where a is the semi-major axis and b is the semi-minor axis. A circle has eccentricity 0. As eccentricity approaches 1 the ellipse becomes more elongated. For a = 5 and b = 3, eccentricity is sqrt(1 minus 9/25) = sqrt(0.64) = 0.80.

Ellipsoid Volume Calculator

What is the formula for the volume of an ellipsoid?

The volume of an ellipsoid is V = (4/3) times pi times a times b times c, where a, b and c are the three semi-axes. If all three semi-axes are equal the formula reduces to the volume of a sphere: (4/3) pi r cubed.

How is ellipsoid surface area calculated?

There is no simple closed-form expression for the surface area of a general ellipsoid. The Knud Thomsen approximation uses SA approximately equal to 4 pi times ((a^p b^p + a^p c^p + b^p c^p) divided by 3)^(1/p) where p is approximately 1.6075. This gives an error of less than 1.061 percent for any ellipsoid.

What is a prolate versus oblate ellipsoid?

A prolate ellipsoid (like a rugby ball) has one semi-axis longer than the other two equal ones. An oblate ellipsoid (like the Earth) is flattened, meaning two equal longer semi-axes and one shorter one. A general ellipsoid has all three semi-axes different.

Equilateral Triangle Calculator

How do you find the area of an equilateral triangle?

Area = (sqrt(3) / 4) x side squared. For side 6: area = (sqrt(3) / 4) x 36 = 9 x sqrt(3) = approximately 15.59 square units. Every interior angle of an equilateral triangle is 60 degrees.

What is the height of an equilateral triangle?

Height = (sqrt(3) / 2) x side. For side 6: height = 3 x sqrt(3) = approximately 5.20 units. The height bisects the base and creates two 30-60-90 right triangles.

What are the inradius and circumradius of an equilateral triangle?

The inradius (largest circle fitting inside) = side / (2 x sqrt(3)). The circumradius (circle through all three corners) = side / sqrt(3). For side 6: inradius = sqrt(3) = 1.73, circumradius = 2 x sqrt(3) = 3.46. The circumradius is always exactly twice the inradius.

Excavation Volume Calculator NZ 2026

What is soil swell or bulking?

When soil is dug it breaks up and takes up more space than it did in the ground. This bulking, or swell, is typically 10 to 15 percent for sand, around 25 percent for common soil, and 40 percent or more for clay and rock. It increases the volume you cart away.

How many truckloads will my excavation be?

Divide the loose volume, after swell, by the truck capacity and round up. A 6 cubic metre tip truck carts away about 6 cubic metres of loose material per trip. This calculator estimates the loads so you can budget cartage and disposal.

How do I estimate disposal weight?

Loose excavated soil weighs roughly 1.4 tonnes per cubic metre, though wet or clay soils are heavier. Transfer stations and cleanfills often charge by weight, so the tonnes estimate here helps you budget disposal, but confirm rates with your site.

Half Angle Calculator

What is the half angle formula for sine?

sin(x/2) = plus or minus the square root of (1 minus cos x) over 2, with the sign set by the quadrant of x over two.

How do I avoid the sign ambiguity?

Work out which quadrant x over two is in, or, as this calculator does, evaluate the function of the actual half angle directly.

What is the tangent half angle form?

tan(x/2) = (1 minus cos x) divided by sin x, which also equals sin x divided by (1 plus cos x).

Hemisphere Volume Calculator

How do you find the volume of a hemisphere?

A hemisphere is half a sphere, so its volume is half the sphere formula: two thirds of pi times the radius cubed. For a radius of 6, that is two thirds times pi times 216, about 452.4 cubic units.

What is the surface area of a hemisphere?

The curved surface is two pi times the radius squared, half of a full sphere. If you also count the flat circular base, the total surface area is three pi times the radius squared. The calculator shows both.

Where do hemisphere calculations come up?

Domes, bowls, tank ends, half-round gauges, igloos and many architectural and engineering shapes are hemispheres or built from them. Knowing the volume helps with capacity and material, and the surface area helps with coating, cladding or heat loss.

Hexagon Calculator

What is the area of a regular hexagon?

The area of a regular hexagon with side length s is (3 times the square root of 3 divided by 2) times s squared, which is approximately 2.5981 times s squared. For example, a hexagon with side 6 has area approximately 2.5981 times 36 = 93.53 square units.

What are the two types of diagonal in a regular hexagon?

A regular hexagon has two distinct diagonal lengths. The long diagonal connects opposite vertices and equals twice the side length (2s). The short diagonal connects vertices separated by one vertex and equals s times the square root of 3, approximately 1.732 times s. For side length 6, the long diagonal is 12 and the short diagonal is approximately 10.39.

What is the apothem of a hexagon?

The apothem of a regular hexagon is the perpendicular distance from the centre to the midpoint of any side. It equals s times the square root of 3 divided by 2, approximately 0.866 times s. For side length 6 the apothem is approximately 5.20. The apothem is also the inradius and equals the circumradius times the cosine of 30 degrees.

Hexagonal Number Calculator

What is a hexagonal number?

A figurate number counting dots in nested hexagons, given by n(2n minus 1): 1, 6, 15, 28 and so on.

Are hexagonal numbers also triangular?

Yes. Every hexagonal number is a triangular number, specifically T(2n minus 1).

What is the 4th hexagonal number?

28, which is also a perfect number.

Hexagonal Prism Volume Calculator

What is the hexagonal prism volume formula?

V = the regular hexagon base area times the prism length. The base area is (3 times the square root of 3, over 2) times the side squared.

Does it assume a regular hexagon?

Yes. It assumes a regular hexagon with six equal sides. An irregular hexagonal cross section needs its base area worked out separately, then multiplied by the length.

What is the base area constant?

Three times the square root of three, divided by two, which is about 2.598076 times the side squared.

Hydraulic Cylinder Force Calculator

Why is the retraction (pull) force lower than the extension (push) force?

During extension, hydraulic pressure acts on the full bore area of the piston. During retraction, the rod takes up some of that area, so pressure acts only on the annular (ring-shaped) area between the bore and the rod. Because the effective area is smaller, the retraction force is always lower than the extension force at the same pressure. A larger rod diameter reduces the annular area further, lowering the pull force more.

What pressure do most hydraulic systems run at?

Light industrial and agricultural hydraulics commonly operate between 140 and 210 bar (2,000 to 3,000 psi). Mobile plant and construction equipment typically runs at 200 to 350 bar. High-pressure systems such as metal presses and injection moulding machines can reach 700 bar or more. System pressure is set by the relief valve, and the cylinder seal ratings must match or exceed this value.

How do I choose the right bore diameter for a required force?

Rearrange the force formula: bore area required = force / pressure. Then bore diameter = 2 x sqrt(area / pi). For example, to achieve 100 kN at 200 bar (20 MPa), the required area is 100,000 / 20,000,000 = 0.005 m², giving a bore diameter of 2 x sqrt(0.005 / pi) = approximately 79.8 mm. Round up to the next standard bore size (typically 80 mm or 90 mm) and confirm the cylinder seals are rated for your working pressure.

Hyperbolic Cosine Calculator

What is the formula for cosh?

cosh(x) = (e^x + e^-x) / 2, built from the exponential function.

Why is cosh never less than one?

Because (e^x + e^-x) / 2 is smallest when x = 0, where it equals 1. It rises on both sides.

What is a catenary?

The curve a hanging cable forms under gravity. Its equation is a scaled cosh, which is why cosh appears in engineering.

Hyperbolic Sine Calculator

What is the formula for sinh?

sinh(x) = (e^x - e^-x) / 2, built from the exponential function. It is not the same as ordinary sine.

Is sinh periodic like sine?

No. Hyperbolic sine is not periodic. It grows without limit as x increases and decreases without limit as x decreases.

What is sinh(0)?

Zero. sinh is an odd function passing through the origin.

Hyperbolic Tangent Calculator

What is the formula for tanh?

tanh(x) = sinh(x) / cosh(x), or equivalently (e^x - e^-x) / (e^x + e^-x).

What range does tanh take?

It always lies strictly between -1 and 1, approaching -1 for large negative x and 1 for large positive x.

Why is tanh used in machine learning?

Its smooth, bounded S shape maps any input to a value between -1 and 1, which is useful as a neuron activation function.

Inverse Hyperbolic Sine Calculator

What is the formula for arcsinh?

arcsinh(x) = ln(x + sqrt(x^2 + 1)), a closed form using the natural logarithm.

What inputs are allowed?

Any real number. Hyperbolic sine covers all real values, so its inverse has no domain restriction.

Is arcsinh the same as 1 / sinh?

No. arcsinh is the inverse function of sinh, not its reciprocal. The reciprocal of sinh is the hyperbolic cosecant.

Inverse Trig Calculator

What is inverse trig used for?

To find an angle when you know a trig ratio. Knowing the opposite and hypotenuse of a right triangle, arcsin gives the angle. From the rise and run of a slope, arctan gives the angle. The inverse functions reverse the ordinary sine, cosine and tangent.

Why won't arcsin or arccos accept my number?

Sine and cosine only ever produce values between minus one and plus one, so their inverses are only defined on that range. A value outside minus one to plus one has no real arcsin or arccos. Arctan by contrast accepts any value.

What is a principal value in inverse trig?

The inverse functions return a single standard angle called the principal value, since many angles share the same ratio. Arctan returns between minus 90 and 90 degrees, arcsin between minus 90 and 90, and arccos between 0 and 180. Consider the problem context for other valid angles.

Isosceles Triangle Calculator

How do you find the area of an isosceles triangle?

Find the height by dropping a perpendicular from the apex to the midpoint of the base. Height = sqrt(leg squared minus (base/2) squared). Area = (1/2) x base x height. For legs 5 and base 6: height = sqrt(25-9) = 4, area = (1/2) x 6 x 4 = 12 square units.

What is the apex angle of an isosceles triangle?

The apex angle is at the vertex between the two equal legs. It equals 2 x arcsin((base/2) / leg). For legs 5 and base 6: apex = 2 x arcsin(3/5) = 2 x 36.87 = 73.74 degrees.

What makes a triangle isosceles?

An isosceles triangle has at least two sides of equal length, called the legs, and a third side called the base. The two base angles (opposite the equal legs) are always equal to each other. An equilateral triangle is a special case where all three sides are equal.

Law of Cosines Calculator

What is the law of cosines?

The law of cosines is the formula c squared = a squared + b squared minus 2ab times cos(C), where a, b and c are the side lengths of any triangle and C is the angle opposite side c. It generalises the Pythagorean theorem: when C is 90 degrees, cos(C) is zero and the formula reduces to c squared = a squared + b squared.

What is the SAS case?

SAS stands for side-angle-side. You know two sides and the angle between them. In this calculator that means you enter sides a and b plus angle C. The law of cosines gives side c directly, and then the remaining angles A and B can be found using the law of cosines again or the law of sines.

What is the SSS case?

SSS stands for side-side-side. You know all three side lengths. The law of cosines can be rearranged to find each angle: cos(C) = (a squared + b squared minus c squared) divided by (2ab). Enter all three sides and the calculator finds all three angles. This calculator uses the SAS mode; for SSS mode enter the three sides and set C to the angle you want to find via trial or use the SSS-specific form.

Law of Sines Calculator

What is the law of sines?

The law of sines states that in any triangle, each side divided by the sine of its opposite angle equals the same constant: a divided by sin(A) equals b divided by sin(B) equals c divided by sin(C). This common ratio is twice the circumradius of the triangle. The law applies to all triangles, not just right-angled ones.

When can I use the law of sines?

The law of sines is useful when you know two angles and any one side (AAS or ASA), or when you know two sides and an angle opposite one of them (SSA). For the AAS and ASA cases the triangle is uniquely determined. The SSA case can sometimes produce two valid triangles (the ambiguous case), which this calculator does not resolve automatically.

What is the circumradius?

The circumradius is the radius of the circle that passes through all three vertices of the triangle (the circumscribed circle or circumcircle). From the law of sines, the circumradius R equals any side divided by twice the sine of its opposite angle: R = a / (2 sin A).

Law of Sines & Cosines Calculator NZ

When do I use the law of cosines versus the law of sines?

Use the law of cosines when you know two sides and the angle between them, or all three sides. Use the law of sines when you have a side and its opposite angle, plus one more piece. This calculator combines both to solve a side-angle-side triangle.

What is side-angle-side?

It means you know two sides of a triangle and the angle between them. This always defines a single, unique triangle, so it is a reliable case to solve, avoiding the ambiguous results that can come from knowing two sides and a non-included angle.

How do I find the area of a non-right triangle?

When you know two sides and the included angle, the area is half the product of those two sides times the sine of the angle between them. This calculator computes it automatically along with the missing side and angles.

nth Root & Cube Root Calculator NZ

How do I calculate the nth root?

The nth root of a number is that number raised to the power of one divided by n. For example, the cube root is the power one third, and the fourth root is the power one quarter. This calculator computes any nth root, plus the square and cube roots.

Can you take the root of a negative number?

Odd roots of negative numbers are fine: the cube root of minus 8 is minus 2. But even roots, such as the square or fourth root of a negative number, are not real numbers, because no real number squared gives a negative. The calculator flags these cases.

What is the cube root used for?

The cube root recovers a length from a volume, since volume is a length cubed. It also appears in scaling problems and some growth-rate calculations. This calculator shows the cube root alongside any nth root you choose.

Parallelogram Calculator

What is the area formula for a parallelogram?

The area of a parallelogram equals the base times the height, where the height is the perpendicular distance between the two parallel bases, not the slant side length. Alternatively, if you know two adjacent sides a and b and the included angle theta, the area equals a times b times sin(theta). These two formulas are equivalent because the height h equals b times sin(theta).

How do you find the height of a parallelogram?

The height of a parallelogram is the perpendicular distance between the base and the opposite side. If you know the slant side s and the included angle theta, the height equals s times sin(theta). This height is different from the side length: a parallelogram with the same base and side can have different areas depending on how tilted it is.

How are the diagonals of a parallelogram calculated?

A parallelogram has two diagonals. Using the law of cosines: the shorter diagonal d1 squared equals a squared plus b squared minus 2ab times cos(theta), and the longer diagonal d2 squared equals a squared plus b squared plus 2ab times cos(theta), where a and b are the adjacent sides and theta is the included angle.

Pascal’s Triangle Calculator

How is Pascal’s triangle built?

Start each row with 1, and every inner number is the sum of the two numbers above it. The ends are always 1.

What is row n?

The binomial coefficients n choose 0 through n choose n, the coefficients of (a + b) to the n.

Why does each row sum to a power of 2?

Because the sum of n choose k over all k equals 2 to the n, the number of subsets of an n element set.

Pentagon Calculator

What is the area formula for a regular pentagon?

The area of a regular pentagon with side length s is (s squared times the square root of 5 times (5 + 2 times the square root of 5)) divided by 4, which simplifies approximately to 1.7205 times s squared. Alternatively you can use area = 5 times s times apothem divided by 2, where the apothem is the perpendicular distance from the centre to the midpoint of a side.

What is the apothem of a pentagon?

The apothem of a regular polygon is the perpendicular distance from its centre to the midpoint of any side. For a regular pentagon with side s, the apothem equals s divided by (2 times tan(36 degrees)), which is approximately 0.6882 times s. The apothem is used in the standard area formula: area equals half the perimeter times the apothem.

What is the diagonal of a regular pentagon?

A regular pentagon has five diagonals, all of equal length. The diagonal length is the golden ratio phi (approximately 1.6180) times the side length s, so diagonal = s times (1 + sqrt(5)) / 2. The golden ratio appears naturally in regular pentagons, making them the basis for the construction of the icosahedron and dodecahedron in 3D geometry.

Pentagonal Prism Volume Calculator

What is the pentagonal prism volume formula?

V = the regular pentagon base area times the prism length. The base area is one quarter times five times the side squared, times the cotangent of pi over five.

Does it assume a regular pentagon?

Yes. It assumes a regular pentagon with five equal sides. An irregular cross section needs its own base area, then multiplied by the length.

What is the base area constant?

About 1.720477 times the side squared for a regular pentagon.

Perimeter Calculator

What is the perimeter of a rectangle?

The perimeter of a rectangle is twice the sum of its length and width: P = 2 times (length + width). The area is length times width. For a rectangle 6 units long and 4 units wide, the perimeter is 2 times (6 + 4) = 20 units and the area is 6 times 4 = 24 square units.

What is the perimeter of a circle?

The perimeter of a circle is called its circumference. It equals 2 times pi times the radius, or equivalently pi times the diameter. For a circle of radius 5, the circumference is 2 times pi times 5 = approximately 31.42 units and the area is pi times 25 = approximately 78.54 square units.

How do you find the perimeter of a regular polygon?

A regular polygon has n sides of equal length s. Its perimeter is simply n times s. The area is (n times s squared) divided by (4 times tan(pi divided by n)). For example, a regular hexagon with side length 4 has a perimeter of 24 and an area of approximately 41.57 square units.

Pipe Volume Calculator

How do you calculate the volume of a pipe?

Volume of a cylindrical pipe equals pi times the inner radius squared times the length: V = pi * r^2 * L. Use the inner radius (half the inner diameter), not the outer radius. The result in cubic metres is converted to litres by multiplying by 1000 (since 1 m3 = 1000 litres).

How long does it take to fill a pipe at a given flow rate?

Fill time in minutes equals volume in litres divided by flow rate in litres per minute. For example, a pipe holding 78.54 litres at a flow rate of 10 L/min takes 78.54 divided by 10, which is about 7.85 minutes to fill. If the flow rate is zero or not entered, the fill time is shown as not applicable.

What is the difference between inner diameter and outer diameter for pipes?

For fluid volume calculations, you must use the inner diameter (bore). The outer diameter includes the pipe wall thickness, which does not carry fluid. Pipe specifications typically list the nominal bore (NB) or the actual inner diameter alongside the outer diameter and wall thickness. Using the outer diameter instead of the inner diameter will overestimate the volume.

Pipe Volume & Flow Calculator NZ 2026

How do I find the volume of water in a pipe?

Treat the pipe as a cylinder: the cross-sectional area is pi times the internal radius squared, and the volume is that area times the length. Multiply cubic metres by 1,000 for litres. This calculator does it from the internal diameter and length you enter.

Should I use internal or external pipe diameter?

Use the internal diameter, the bore the water flows through. Pipe is often sold by a nominal or outside size that is larger than the bore, so using the outside diameter overstates both the volume and the flow. Check the pipe's specifications for the internal size.

What is a normal water flow velocity?

Water in household and irrigation pipes commonly flows at around 1 to 2 metres per second. Lower speeds reduce noise, friction loss and wear, while higher speeds can cause water hammer and erosion. Designers pick a velocity to balance pipe size against pressure loss.

Pyramid Volume and Surface Area Calculator

What is the formula for the volume of a pyramid?

The volume of any pyramid is one third of the base area multiplied by the perpendicular height: V = (1/3) x Base Area x h. For a square-base pyramid with side a and height h, this gives V = (1/3) x a^2 x h. For a rectangular base with length l and width w, it is V = (1/3) x l x w x h. For a triangular base, the base area is (1/2) x base x height of the triangle.

How do you find the slant height of a pyramid?

The slant height (l) runs from the apex down to the midpoint of a base edge. For a square-base pyramid with base side a and perpendicular height h, slant height = sqrt(h^2 + (a/2)^2). For a rectangular pyramid, there are two different slant heights: one for the length side, sqrt(h^2 + (w/2)^2), and one for the width side, sqrt(h^2 + (l/2)^2). The slant height is needed to calculate the lateral surface area.

What is the difference between lateral surface area and total surface area of a pyramid?

Lateral surface area is the combined area of the triangular faces only, not including the base. Total surface area adds the base area to the lateral area. For a square-base pyramid: lateral area = 2 x a x slant height, and total surface area = a^2 + 2 x a x slant height. If you are painting the outside of a solid pyramid (base included), use total surface area. If you are calculating just the sloping faces (for example, for a tent or a roof), use lateral surface area.

Pythagorean Theorem Calculator

What is the Pythagorean theorem?

For a right angled triangle, the square of the longest side, the hypotenuse, equals the sum of the squares of the other two sides. Written as a squared plus b squared equals c squared, it lets you find any side from the other two.

How do I find the hypotenuse?

Square the two shorter sides, add them, and take the square root. So with sides 3 and 4, the hypotenuse is the square root of 9 plus 16, which is 5. Enter the two sides and the calculator finds it.

Can I find a shorter side instead?

Yes. If you know the hypotenuse and one short side, subtract the square of the short side from the square of the hypotenuse and take the square root. The calculator solves for whichever side you leave blank.

Rectangular Box (Cuboid) Calculator

What is the formula for the volume of a rectangular box?

The volume of a rectangular box (cuboid) is V = l x w x h, where l is the length, w is the width, and h is the height. All three dimensions must be in the same unit. For example, a box measuring 5 m x 4 m x 3 m has a volume of 60 cubic metres.

How do you calculate the surface area of a cuboid?

The total surface area of a cuboid is SA = 2(lw + lh + wh). A cuboid has six rectangular faces: two faces of l x w, two faces of l x h, and two faces of w x h. For a 5 x 4 x 3 box: SA = 2(20 + 15 + 12) = 2 x 47 = 94 square units.

What is the space diagonal of a rectangular box?

The space diagonal runs from one corner of the box to the opposite corner through the interior. Its length is d = sqrt(l^2 + w^2 + h^2). For a 5 x 4 x 3 box: d = sqrt(25 + 16 + 9) = sqrt(50) = approximately 7.071 units. Each face also has its own face diagonal, calculated as the square root of the sum of squares of its two side lengths.

Rectangular Pyramid Volume Calculator

What is the formula for a rectangular pyramid volume?

V = one third times base length times base width times height. The base area is length times width, then take a third of that times the height.

Is height the slant length or the vertical height?

Use the vertical (perpendicular) height from the base to the apex, not the slanted face or edge length.

Why one third?

A pyramid fills exactly one third of the prism or box with the same base and height. This holds for any pyramid, whatever the base shape.

Reference Angle Calculator

What is a reference angle?

A reference angle is the acute angle between 0 and 90 degrees formed between the terminal side of an angle and the x-axis. It lets you work out the trig functions of any angle from the values of a familiar acute angle, adjusting the sign for the quadrant.

How do you find the reference angle for 150 degrees?

150 degrees is in the second quadrant, between 90 and 180 degrees. The reference angle is 180 minus 150 = 30 degrees. Sine, cosine and tangent of 150 degrees have the same size as those of 30 degrees, with signs set by the second quadrant.

How do you find reference angles in general?

First reduce the angle to between 0 and 360 degrees. In quadrant 1 the reference angle is the angle itself. In quadrant 2 it is 180 minus the angle. In quadrant 3 it is the angle minus 180. In quadrant 4 it is 360 minus the angle.

Regular Decagon Area Calculator

What is the area formula for a regular decagon?

A = one quarter times ten times the side length squared, times the cotangent of pi over 10. The calculator evaluates the cotangent for you.

Does this work only for a regular decagon?

Yes. It assumes a regular decagon, meaning all ten sides and all angles are equal. An irregular decagon needs a different method, such as splitting it into triangles.

What units does it use?

Whatever unit you enter. A side in metres gives an area in square metres, a side in centimetres gives square centimetres. Keep the side in one unit.

Regular Decagon Perimeter Calculator

How do you find the perimeter of a regular decagon?

Multiply the side length by ten. Every side of a regular decagon is the same length, so P = 10 times the side.

Is the perimeter the same as the circumference?

For a polygon we call the distance around the outside the perimeter. Circumference is the same idea but is the word used for a circle.

What if the decagon is irregular?

Then the sides differ, so add the individual side lengths instead. This calculator assumes a regular decagon with equal sides.

Regular Heptagon Area Calculator

What is the area formula for a regular heptagon?

A = one quarter times seven times the side length squared, times the cotangent of pi over 7. The calculator evaluates the cotangent for you.

Does this work only for a regular heptagon?

Yes. It assumes a regular heptagon, meaning all seven sides and all angles are equal. An irregular heptagon needs a different method, such as splitting it into triangles.

What units does it use?

Whatever unit you enter. A side in metres gives an area in square metres, a side in centimetres gives square centimetres. Keep the side in one unit.

Regular Heptagon Perimeter Calculator

How do you find the perimeter of a regular heptagon?

Multiply the side length by seven. Every side of a regular heptagon is the same length, so P = 7 times the side.

Is the perimeter the same as the circumference?

For a polygon we call the distance around the outside the perimeter. Circumference is the same idea but is the word used for a circle.

What if the heptagon is irregular?

Then the sides differ, so add the individual side lengths instead. This calculator assumes a regular heptagon with equal sides.

Regular Hexagon Area Calculator

What is the area formula for a regular hexagon?

A = one quarter times six times the side length squared, times the cotangent of pi over 6. The calculator evaluates the cotangent for you.

Does this work only for a regular hexagon?

Yes. It assumes a regular hexagon, meaning all six sides and all angles are equal. An irregular hexagon needs a different method, such as splitting it into triangles.

What units does it use?

Whatever unit you enter. A side in metres gives an area in square metres, a side in centimetres gives square centimetres. Keep the side in one unit.

Regular Hexagon Perimeter Calculator

How do you find the perimeter of a regular hexagon?

Multiply the side length by six. Every side of a regular hexagon is the same length, so P = 6 times the side.

Is the perimeter the same as the circumference?

For a polygon we call the distance around the outside the perimeter. Circumference is the same idea but is the word used for a circle.

What if the hexagon is irregular?

Then the sides differ, so add the individual side lengths instead. This calculator assumes a regular hexagon with equal sides.

Regular Nonagon Area Calculator

What is the area formula for a regular nonagon?

A = one quarter times nine times the side length squared, times the cotangent of pi over 9. The calculator evaluates the cotangent for you.

Does this work only for a regular nonagon?

Yes. It assumes a regular nonagon, meaning all nine sides and all angles are equal. An irregular nonagon needs a different method, such as splitting it into triangles.

What units does it use?

Whatever unit you enter. A side in metres gives an area in square metres, a side in centimetres gives square centimetres. Keep the side in one unit.

Regular Nonagon Perimeter Calculator

How do you find the perimeter of a regular nonagon?

Multiply the side length by nine. Every side of a regular nonagon is the same length, so P = 9 times the side.

Is the perimeter the same as the circumference?

For a polygon we call the distance around the outside the perimeter. Circumference is the same idea but is the word used for a circle.

What if the nonagon is irregular?

Then the sides differ, so add the individual side lengths instead. This calculator assumes a regular nonagon with equal sides.

Regular Octagon Area Calculator

What is the area formula for a regular octagon?

A = one quarter times eight times the side length squared, times the cotangent of pi over 8. The calculator evaluates the cotangent for you.

Does this work only for a regular octagon?

Yes. It assumes a regular octagon, meaning all eight sides and all angles are equal. An irregular octagon needs a different method, such as splitting it into triangles.

What units does it use?

Whatever unit you enter. A side in metres gives an area in square metres, a side in centimetres gives square centimetres. Keep the side in one unit.

Regular Octagon Perimeter Calculator

How do you find the perimeter of a regular octagon?

Multiply the side length by eight. Every side of a regular octagon is the same length, so P = 8 times the side.

Is the perimeter the same as the circumference?

For a polygon we call the distance around the outside the perimeter. Circumference is the same idea but is the word used for a circle.

What if the octagon is irregular?

Then the sides differ, so add the individual side lengths instead. This calculator assumes a regular octagon with equal sides.

Regular Pentagon Area Calculator

What is the area formula for a regular pentagon?

A = one quarter times five times the side length squared, times the cotangent of pi over 5. The calculator evaluates the cotangent for you.

Does this work only for a regular pentagon?

Yes. It assumes a regular pentagon, meaning all five sides and all angles are equal. An irregular pentagon needs a different method, such as splitting it into triangles.

What units does it use?

Whatever unit you enter. A side in metres gives an area in square metres, a side in centimetres gives square centimetres. Keep the side in one unit.

Regular Pentagon Perimeter Calculator

How do you find the perimeter of a regular pentagon?

Multiply the side length by five. Every side of a regular pentagon is the same length, so P = 5 times the side.

Is the perimeter the same as the circumference?

For a polygon we call the distance around the outside the perimeter. Circumference is the same idea but is the word used for a circle.

What if the pentagon is irregular?

Then the sides differ, so add the individual side lengths instead. This calculator assumes a regular pentagon with equal sides.

Regular Polygon Calculator

What is a regular polygon?

A regular polygon is a flat shape with equal side lengths and equal interior angles. Examples include the equilateral triangle (3 sides), square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), hendecagon (11), and dodecagon (12). All vertices lie on a common circle called the circumscribed circle.

How do you calculate the interior angle of a regular polygon?

The interior angle of a regular polygon with n sides is (n minus 2) times 180 degrees divided by n. For a hexagon (n = 6): (6 minus 2) times 180 divided by 6 = 4 times 180 divided by 6 = 120 degrees. The exterior angle is 360 divided by n.

What is the apothem of a regular polygon?

The apothem is the perpendicular distance from the centre of the polygon to the midpoint of any side. It equals half the side length divided by the tangent of pi divided by n, or equivalently s divided by (2 times tan(pi/n)). The apothem is used to calculate the area: area = 0.5 times perimeter times apothem.

Rhombus Calculator

What is a rhombus?

A rhombus is a parallelogram with all four sides equal in length. Its opposite angles are equal and its adjacent angles add up to 180 degrees. The two diagonals of a rhombus bisect each other at right angles, which is the key property that distinguishes a rhombus from a general parallelogram. A square is a special rhombus where all angles are 90 degrees.

What is the area formula for a rhombus?

The area of a rhombus equals the side squared times sin(theta), where theta is one of the interior angles. It can also be calculated as half the product of the two diagonals: area = d1 times d2 divided by 2. Both formulas give the same result. The sine formula is convenient when you know the side and angle; the diagonal formula is convenient when you have measured both diagonals directly.

How do you find the diagonals of a rhombus?

If the rhombus has side a and one interior angle theta, the two diagonals are d1 = 2a times sin(theta/2) and d2 = 2a times cos(theta/2). These follow from the fact that the diagonals bisect the angles of the rhombus and meet at right angles, forming four right-angled triangles inside the shape.

Right Angle Triangle Calculator

What is the Right Angle Triangle Calculator?

Calculate missing sides, angles, and area of any right-angle triangle. Enter any two known sides to find the hypotenuse using the Pythagorean theorem, plus angle A, angle B, and triangle area.

Is the Right Angle Triangle Calculator free to use?

Yes. The Right Angle Triangle Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

Is the Right Angle Triangle Calculator made for New Zealand?

Yes. It is hosted on Calculate.co.nz for New Zealand students and tradespeople. The calculations use the Pythagorean theorem and standard trigonometry, which work the same way anywhere, so no country-specific rules are involved.

Sine Calculator

How do I find the sine of an angle?

Enter the angle and pick degrees or radians. The calculator converts to radians and evaluates the sine.

Degrees or radians?

Either. Choose the unit in the dropdown. The angle is shown in both units beside the result so you can check.

Why is the result the same every 360 degrees?

The trig functions are periodic. Sine and cosine repeat every 360 degrees (2 pi radians), so equivalent angles give the same value.

Sine Cosine Tangent Calculator

What are sine, cosine and tangent?

They are the three main trigonometric ratios of a right triangle. Sine is the opposite side over the hypotenuse, cosine is the adjacent over the hypotenuse, and tangent is the opposite over the adjacent, remembered as SOH CAH TOA.

What are the inverse functions for?

Inverse sine, cosine and tangent, written as arcsin, arccos and arctan, find the angle that produces a given ratio. They answer the question, which angle has this sine?

Why is tangent undefined at 90 degrees?

Tangent is sine divided by cosine, and cosine is zero at 90 and 270 degrees, so dividing by zero leaves tangent undefined at those angles.

Sine Rule Ambiguous Case Calculator

Why is SSA called the ambiguous case?

Because the known sine can correspond to two angles, an acute one and its obtuse supplement, so there can be two different triangles, or one, or none.

When is there no triangle?

When b sin A / a is greater than 1, the sine rule has no solution and no triangle can be formed.

When are there two triangles?

When both the acute angle B1 and its supplement B2 keep the angle sum A + B below 180 degrees.

Small Angle Approximation Calculator

When is the small angle approximation valid?

For small angles in radians. The smaller the angle, the better. Below about 0.24 radians (14 degrees) the error in sin x approx x is under one percent.

Must the angle be in radians?

Yes. The approximation sin x approx x only holds when x is in radians, not degrees.

What is the cosine approximation?

cos x approx 1 - x squared over two for small x in radians.

Sphere Surface Area Calculator

What is the formula for the surface area of a sphere?

Surface area = 4 x pi x r squared. For a sphere of radius 6: SA = 4 x pi x 36 = 452.39 square units. This is equivalent to the area of exactly four great circles, each with area pi x r squared.

What is the volume of a sphere?

Volume = (4/3) x pi x r cubed. For r=6: volume = (4/3) x pi x 216 = 904.78 cubic units. The ratio of surface area to volume decreases as the sphere grows, which is why large spheres lose heat more slowly relative to their mass.

What is a great circle of a sphere?

A great circle is the largest possible circle you can draw on a sphere, obtained by slicing through the exact centre. Its radius equals the sphere's radius. The great circle area is pi x r squared. A sphere's total surface area is exactly four times this great circle area.

Sphere Volume Calculator

What is the volume of a sphere?

The volume of a sphere is four thirds times pi times the radius cubed. For a radius of 5 that is about 523.6 cubic units. Enter the radius and the calculator works it out.

What is the surface area of a sphere?

The surface area is four times pi times the radius squared. For a radius of 5 it is about 314.2 square units. The calculator shows it alongside the volume.

How do I find the radius from the diameter?

The radius is half the diameter. So if you know the distance across the ball through its centre, divide by two and enter that as the radius.

Square Calculator

What is the formula for the area of a square?

Area = side squared (s x s). For a square with side 7: area = 7 x 7 = 49 square units.

What is the diagonal of a square?

The diagonal of a square runs from one corner to the opposite corner. Its length is the side times the square root of 2. For side 7: diagonal = 7 x sqrt(2) = approximately 9.90 units. This comes from applying Pythagoras to the two sides of the right triangle formed by the diagonal.

How do you find the perimeter of a square?

Perimeter = 4 x side. For a square with side 7: perimeter = 4 x 7 = 28 units. All four sides of a square are equal, so the perimeter is simply four times the side length.

Square Root Calculator

What is a square root?

The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 times 5 is 25. Most numbers do not have a whole square root, so the result is a decimal.

Can you take the square root of a negative number?

Not within the real numbers, because no real number multiplied by itself gives a negative result. The square root of a negative number is an imaginary number, which is beyond this calculator. Enter a number of zero or more.

What is a perfect square?

A perfect square is a whole number whose square root is also a whole number, such as 4, 9, 16, and 25. Their roots are 2, 3, 4, and 5. Numbers in between have square roots that are decimals.

STP Gas Volume Calculator

What is the molar volume of a gas?

The molar volume is the volume one mole of any ideal gas occupies. At STP, 0 degrees and 1 atmosphere, it is about 22.4 litres per mole. At RTP, around 25 degrees, the common school value is 24.0 litres per mole.

How do you find the volume of a gas from moles?

Multiply the number of moles by the molar volume. At STP, 2 moles of gas occupy 2 times 22.4, which is 44.8 litres. To find moles from a volume, divide the volume by the molar volume.

Does the type of gas matter?

For an ideal gas, no. Equal numbers of moles of any gas take up the same volume at the same temperature and pressure, which is Avogadro's law. Real gases follow this closely under everyday conditions.

Surface Area Calculator (Multi-Solid)

What is the surface area of a sphere?

The surface area of a sphere is 4 x pi x r^2, where r is the radius. For example, a sphere with a radius of 5 cm has a surface area of 4 x 3.14159 x 25 = 314.16 cm squared. This formula gives the total area of the curved surface that encloses the sphere.

What is the difference between total surface area and lateral surface area?

Total surface area (TSA) is the combined area of every face of a solid, including the top, bottom, and all sides. Lateral surface area (LSA) includes only the side faces, excluding the top and bottom bases. For a cylinder, the lateral surface area is 2 x pi x r x h (just the curved wall), while the total surface area adds two circular bases: TSA = 2 x pi x r x (r + h). This calculator shows total surface area for all solids.

How do I find the surface area of a cone?

The total surface area of a cone is pi x r x (r + l), where r is the base radius and l is the slant height. If you know the height (h) instead of the slant height, first calculate l = sqrt(r^2 + h^2). For example, a cone with base radius 3 cm and height 4 cm has slant height sqrt(9 + 16) = 5 cm, and a total surface area of pi x 3 x (3 + 5) = 3.14159 x 24 = 75.40 cm squared.

Surface Area Calculator

What is surface area?

Surface area is the total area of all the outside faces of a three-dimensional object. It is measured in square units and tells you how much material it would take to cover the object, how much paint or coating you need, or how much surface is available for heat to escape.

How do you find the surface area of a cylinder?

Add the two circular ends to the curved side. Each end is pi x r squared, and the curved side unrolls into a rectangle of 2 x pi x r times the height. Total SA = 2 x pi x r x (r + h).

What is the difference between surface area and volume?

Surface area measures the outside of an object in square units, while volume measures the space inside in cubic units. You use surface area for covering, coating or wrapping, and volume for capacity or how much it holds.

Surface Area to Volume Ratio Calculator

Why does surface area to volume ratio matter in biology?

Cells and organisms exchange nutrients, gases and heat through their surfaces but use energy in proportion to their volume. A high SA:V ratio means efficient exchange relative to metabolic demand. As a cell grows larger its volume increases faster than its surface, so the SA:V ratio falls. Beyond a critical size, diffusion alone cannot supply enough nutrients, which is why large organisms need specialised transport systems like blood vessels and lungs.

What is the SA:V ratio of a cube with side 5?

A cube with side length s = 5 has surface area 6 times 5 squared = 150 square units and volume 5 cubed = 125 cubic units. The SA:V ratio is 150 divided by 125 = 1.2 per unit. For a cube the ratio simplifies to 6 divided by s, so doubling the side length halves the ratio.

Which shape has the lowest SA:V ratio for a given volume?

A sphere has the lowest possible SA:V ratio for any given volume. The sphere SA:V ratio is 3 divided by r (radius). A cube with the same volume has a higher ratio because the sphere is the most compact shape possible, minimising surface for a given enclosed volume.

Tangent Half Angle Calculator

What is the Weierstrass substitution?

Setting t = tan(x/2), so that sin x = 2t/(1+t squared) and cos x = (1 minus t squared)/(1 + t squared). It turns trig integrals into algebraic ones.

Why is it useful?

It converts an integral with sin x and cos x into an integral of a rational function of t, which can be solved with partial fractions.

When does t blow up?

When x is an odd multiple of 180 degrees, tan(x/2) is undefined; the substitution is used away from those points.

Tangent Line Calculator

What is a tangent line?

A tangent line to a curve at a point is the straight line that just touches the curve at that point and has the same slope as the curve at that instant. The slope of the tangent is the derivative of the function evaluated at the point. Tangent lines are used to approximate functions locally, to find the direction of motion, and as the starting point for linear approximations.

How do you find the equation of a tangent line?

To find the tangent at x = a: first evaluate f(a) to get the y coordinate of the point. Then find the derivative f'(a), which is the slope of the tangent. Finally, use the point-slope form y minus f(a) equals f'(a) times (x minus a), rearranging to y = f'(a) times x plus (f(a) minus f'(a) times a).

What is the normal line?

The normal line at a point is perpendicular to the tangent line at that same point. If the tangent has slope m, the normal has slope -1/m (the negative reciprocal). The normal line equation uses the same point (a, f(a)) but with the perpendicular slope. If the tangent is horizontal (slope 0), the normal is a vertical line x = a.

Tank Volume Calculator

How do you calculate tank volume?

For a cylinder, multiply the circular cross-section area, pi over 4 times the diameter squared, by the length or height. For a rectangular tank, multiply length by width by height. Multiply cubic metres by 1,000 to get litres.

How do you find the volume in a partly full tank?

For vertical cylinder and rectangular tanks the fill is in direct proportion to the depth. For a horizontal cylinder it is not: the calculator uses the area of the circular segment filled at that depth, multiplied by the tank length, which is why a half-depth horizontal tank is exactly half full.

What units should I use?

Enter all dimensions in metres and the calculator returns the volume in cubic metres and litres, since one cubic metre is 1,000 litres. Keep every measurement in the same unit so the result is correct.

Torus Volume Calculator

What is a torus?

A torus is the three-dimensional solid formed by rotating a circle around an axis in its own plane that does not intersect the circle. It looks like a donut or an inflatable ring. It is defined by two radii: R is the major radius from the centre of the torus to the centre of the tube, and r is the minor radius (tube radius), the radius of the circular cross-section.

What is the volume formula for a torus?

The volume of a torus is 2 pi squared times R times r squared, where R is the major radius and r is the tube radius. This formula comes from Pappus's centroid theorem: the volume of a solid of revolution equals the area of the cross-section (pi r squared) times the distance the centroid travels (2 pi R).

What is the surface area formula for a torus?

The surface area of a torus is 4 pi squared times R times r. Again this follows from Pappus's theorem: the surface area equals the perimeter of the cross-section circle (2 pi r) times the distance the centroid travels (2 pi R), giving 4 pi squared R r. For the default values R = 5 and r = 2, both volume and surface area equal approximately 394.78.

Trapezoid Calculator

What is the formula for the area of a trapezoid?

The area of a trapezoid is half the sum of the two parallel sides multiplied by the perpendicular height between them: Area = 0.5 x (a + b) x h. This is equivalent to multiplying the median (midsegment) by the height.

What is the median (midsegment) of a trapezoid?

The median, also called the midsegment, is the line segment connecting the midpoints of the two non-parallel sides (legs). Its length equals the average of the two parallel sides: median = (a + b) / 2. Multiplying the median by the height gives the area.

Is a trapezoid the same as a trapezium?

In most modern usage, trapezoid and trapezium refer to the same four-sided shape that has exactly one pair of parallel sides. The area, perimeter, and median formulas are identical regardless of which name is used.

Triangle Angle Calculator

Why do the angles of a triangle always add up to 180 degrees?

In Euclidean (flat) geometry, the interior angles of any triangle sum to exactly 180 degrees. This can be shown by drawing a line through one vertex parallel to the opposite side: the alternate interior angles formed equal the two base angles, and the three angles along the straight line sum to 180 degrees. This holds for all triangles regardless of shape or size.

What is an acute triangle?

An acute triangle has all three interior angles strictly less than 90 degrees. An obtuse triangle has one angle greater than 90 degrees. A right triangle has exactly one angle equal to 90 degrees. An equilateral triangle is a special case of an acute triangle where all three angles are exactly 60 degrees.

Can a triangle have two obtuse angles?

No. Two obtuse angles would sum to more than 180 degrees, leaving no room for a positive third angle. A valid triangle must have three positive angles that sum to exactly 180 degrees, so it can contain at most one obtuse angle and at most one right angle.

Triangle Calculator (General)

What is the law of cosines and when is it used?

The law of cosines states that c² = a² + b² - 2ab·cos(C), where a, b, and c are the sides of a triangle and C is the angle opposite side c. It is used when you know all three sides (SSS) or two sides and the included angle (SAS). In the SSS case it lets you find all angles. In the SAS case it finds the unknown side, after which the law of sines resolves the remaining angles. The law of cosines is a generalisation of Pythagoras's theorem: when C = 90°, cos(90°) = 0 and the formula reduces to c² = a² + b².

What is the ambiguous case (SSA) in triangle solving?

The ambiguous case occurs when you know two sides and an angle that is not between those two sides (SSA or the angle-side-side case). Given side a, side b, and angle A (opposite to a), there may be zero, one, or two valid triangles. If a is shorter than the perpendicular height h = b·sin(A), no triangle exists. If a equals h, exactly one right triangle exists. If h is less than a and a is less than b, two different triangles satisfy the given information. If a is greater than or equal to b, exactly one triangle exists. This calculator checks all cases and reports whether a second solution exists.

How do I calculate the area of a triangle from sides and angles?

The most direct formula when you know two sides and the included angle is: Area = (1/2) × a × b × sin(C). For example, with sides of 5 and 7 and an included angle of 60°, the area is 0.5 × 5 × 7 × sin(60°) = 15.16 square units. When you know all three sides you can use Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. Both formulas give the same result once all sides and angles are known.

Triangle Solver

What does SSS, SAS, ASA and AAS mean for triangles?

These abbreviations describe which three values you know. SSS means all three sides. SAS means two sides and the angle between them. ASA means two angles and the included side. AAS means two angles and a non-included side. Any combination fully determines the triangle.

How do you find the area of a triangle from its sides?

Use Heron's formula: calculate the semi-perimeter s = (a + b + c) / 2, then area = square root of s(s-a)(s-b)(s-c). If you know two sides and the included angle, use area = 0.5 times a times b times sin(C).

What is the law of cosines?

The law of cosines states c squared = a squared + b squared minus 2ab times cos(C). It generalises Pythagoras for any triangle and lets you find a side or angle when you know the other three values.

Triangular Prism Calculator

How do you find the volume of a triangular prism?

Find the area of the triangular cross-section, then multiply by the length of the prism. Volume = triangle area x length. For a 3-4-5 right triangle of area 6 and a prism 10 units long, the volume is 60 cubic units.

How do you find the surface area of a triangular prism?

Add the two triangular ends to the three rectangular sides. Surface area = 2 x triangle area + perimeter of triangle x length. For a 3-4-5 triangle with perimeter 12 and length 10: SA = 2 x 6 + 12 x 10 = 132 square units.

What is Heron's formula?

Heron's formula gives the area of any triangle from its three side lengths alone. With semi-perimeter s = (a + b + c) / 2, the area = sqrt(s(s-a)(s-b)(s-c)). It works for any valid triangle without needing the height.

Triangular Pyramid Volume Calculator

What is the triangular pyramid volume formula?

V = one third times the base triangle area times the pyramid height, and the base triangle area is one half its base times its height.

Is this the same as a tetrahedron?

A tetrahedron is a triangular pyramid. This calculator takes the base triangle and the vertical height directly, which works for any triangular pyramid, regular or not.

Which height do I use?

Two heights are involved: the height of the base triangle, and the vertical height of the pyramid from base to apex. Enter both where asked.

Trig Calculator

What does the trig calculator give me?

Enter an angle and it returns the six trigonometric functions: sine, cosine and tangent, plus their reciprocals cosecant, secant and cotangent. You can work in degrees or radians.

Why is tangent sometimes undefined?

Tangent is sine divided by cosine, so it is undefined where cosine is zero, such as at 90 and 270 degrees. The same applies to secant. Cosecant and cotangent are undefined where sine is zero, such as at 0 and 180 degrees.

Should I use degrees or radians?

Degrees are common in everyday geometry and navigation, while radians are standard in calculus and higher maths. Use the toggle to match your problem; the calculator converts internally either way.

Trig Equation Solver

Why does a trig equation have infinitely many solutions?

Because sine, cosine and tangent are periodic, the same value repeats every 360 or 180 degrees, so each base solution generates a family with an integer n.

What is the general solution for tan x = k?

x = arctan(k) + 180 degrees times n, for any integer n, since the tangent repeats every 180 degrees.

Why does sin x = k sometimes say no solution?

Sine and cosine only take values between minus one and one. If k is outside that range there is no real solution.

Trig Identity Calculator

What are the three Pythagorean trig identities?

The three Pythagorean identities are: sin squared theta plus cos squared theta equals 1; 1 plus tan squared theta equals sec squared theta; and 1 plus cot squared theta equals csc squared theta. All three follow from the unit circle definition of sine and cosine and apply for all angles where the functions are defined.

What is the double angle identity for sin(2theta)?

The double angle identity for sine is sin(2 theta) = 2 sin(theta) cos(theta). For cosine there are three equivalent forms: cos(2 theta) = cos squared theta minus sin squared theta = 2 cos squared theta minus 1 = 1 minus 2 sin squared theta. All three cosine forms are equivalent and can be derived from the Pythagorean identity.

How do you verify a trig identity numerically?

To verify an identity numerically, substitute a specific angle and check that the left-hand side equals the right-hand side to many decimal places. For example, for theta = 45 degrees: sin squared 45 + cos squared 45 = 0.5 + 0.5 = 1. While numerical checking confirms an identity for that angle, a formal algebraic proof is needed to establish it for all angles.

Trig Identity Solver

What are the Pythagorean trig identities?

There are three Pythagorean identities that hold for every angle: sin²(θ) + cos²(θ) = 1, 1 + tan²(θ) = sec²(θ), and 1 + cot²(θ) = csc²(θ). These follow from the Pythagorean theorem applied to a unit circle. The first is the most commonly used and can be rearranged to sin²(θ) = 1 − cos²(θ) or cos²(θ) = 1 − sin²(θ).

What are the double angle formulas in trigonometry?

The double angle formulas express the trig functions of 2θ in terms of functions of θ. The main ones are: sin(2θ) = 2 sin(θ) cos(θ); cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ); and tan(2θ) = 2tan(θ) / (1 − tan²(θ)). These are derived by applying the angle addition formulas with both angles equal to θ.

How do half angle formulas work?

The half angle formulas express trig functions of θ/2 in terms of functions of θ. They are: sin(θ/2) = ±√((1 − cos θ) / 2); cos(θ/2) = ±√((1 + cos θ) / 2); and tan(θ/2) = ±√((1 − cos θ) / (1 + cos θ)) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ. The sign depends on which quadrant θ/2 falls in. Half angle formulas are derived from the double angle cosine identity by solving for cos²(θ) and sin²(θ).

Trigonometry Calculator NZ

What is the Trigonometry Calculator NZ?

Free trigonometry calculator for NZ students. Solve right triangles and any triangle given any combination of sides and angles. Calculate sin, cos, tan, plus interactive unit circle and exact values reference. NCEA Levels 1-3.

Is the Trigonometry Calculator NZ free to use?

Yes. The Trigonometry Calculator NZ is free to use on Calculate.co.nz, with no sign-up, paywall or account required.

How does the trigonometry calculator solve a triangle?

It uses the standard trigonometric ratios (sin, cos and tan) together with the law of sines and the law of cosines to find the missing sides and angles from the values you enter. These formulas are the same everywhere in mathematics, so the results do not depend on where you are. The tool also shows exact values and an interactive unit circle to support NCEA Levels 1 to 3.

Unit Circle Calculator

What is the unit circle?

The unit circle is a circle of radius 1 centred at the origin. For any angle measured from the positive x-axis, the point where the angle's ray meets the circle has coordinates equal to the cosine and the sine of that angle. This is why the unit circle underpins all of trigonometry.

How do you find coordinates on the unit circle?

The x-coordinate is the cosine of the angle and the y-coordinate is the sine of the angle. At 30 degrees the point is (cos 30, sin 30) = (0.866, 0.5). The tangent is the y-coordinate divided by the x-coordinate.

What are the 16 standard angles on the unit circle?

The 16 standard angles are 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315 and 330 degrees. These correspond to multiples of 30 and 45 degrees, giving exact values for sine and cosine that are used throughout trigonometry.

Volume Calculator (Multi-Solid)

What is the formula for the volume of a sphere?

The volume of a sphere is V = (4/3) x pi x r^3, where r is the radius. For a sphere with a radius of 5 cm, V = (4/3) x 3.14159 x 125 = 523.60 cm3. The formula grows quickly because volume scales with the cube of the radius: doubling the radius increases the volume by a factor of 8.

How do I find the volume of a cylinder?

The volume of a cylinder is V = pi x r^2 x h, where r is the radius of the circular base and h is the height (or length) of the cylinder. For example, a cylinder with a radius of 3 cm and a height of 10 cm has a volume of 3.14159 x 9 x 10 = 282.74 cm3. If you know the diameter rather than the radius, halve it first.

Why is the volume of a cone one-third of a cylinder?

The formula for a cone is V = (1/3) x pi x r^2 x h, which is exactly one-third of the volume of a cylinder with the same base and height. This relationship is a geometric fact that can be proven by calculus (integration) or by physical demonstration: it takes exactly three cone-fills of water to fill a cylinder of the same base radius and height. The same one-third relationship holds between a rectangular pyramid and a rectangular prism.

Volume Calculator

How do you calculate the volume of a cylinder?

The volume of a cylinder is pi times the radius squared times the height: V = pi r squared h. The total surface area includes the two circular ends plus the curved lateral surface: SA = 2 pi r squared + 2 pi r h.

What is the formula for the volume of a cone?

The volume of a cone is one third of pi times the radius squared times the height: V = (1/3) pi r squared h. A cone has exactly one third the volume of a cylinder with the same base and height.

How is the surface area of a sphere calculated?

The surface area of a sphere is 4 times pi times the radius squared: SA = 4 pi r squared. The volume is (4/3) pi r cubed.

Volume of a Cube Calculator

What is the formula for the volume of a cube?

The volume of a cube is V = s³, where s is the length of one side (all sides of a cube are equal). For example, a cube with a side length of 5 cm has a volume of 5³ = 125 cm³. Volume is always expressed in cubic units (cm³, m³, mm³, in³, ft³).

How is a cube different from a rectangular box (cuboid)?

A cube is a special case of a rectangular box (cuboid) where all three dimensions are equal: length, width, and height are the same. Because of this, the volume formula simplifies from V = l × w × h to V = s × s × s = s³. All six faces of a cube are identical squares, and all twelve edges are the same length.

What is the space diagonal of a cube?

The space diagonal of a cube is the longest diagonal, running from one corner of the cube to the opposite corner through the interior. Its length is s × √3, where s is the side length. For a cube with side 5 cm, the space diagonal is 5 × √3 ≈ 8.66 cm. The face diagonal (across one square face) is shorter, at s × √2 ≈ 7.07 cm for a 5 cm cube.