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.

45-45-90 Triangle Calculator

This calculator finds every measurement of a 45-45-90 triangle, a right isosceles triangle with two 45 degree angles and one 90 degree angle, from a single known value. Because both legs are always equal and the hypotenuse is always exactly the leg length multiplied by the square root of 2, knowing just one side, or even the area or the perimeter, is enough to work out everything else. Choose what you know from the dropdown, leg length, hypotenuse, area, or perimeter, enter that one figure, and the calculator instantly returns the leg, hypotenuse, area, and perimeter, plus the height to the hypotenuse, inradius, and circumradius in a full measurements table. A worked line underneath shows the exact formula used for your chosen input, so you can see how each figure is derived rather than just reading off a number. All calculations use the full precision of the square root of 2 rather than a rounded decimal, so results stay accurate even when you are chaining several conversions together. This is handy for geometry homework, checking a diagonal cut, laying out a mitred joint, or framing a roof at a 45 degree pitch, where the classic root two ratio turns up again and again. Because the shape has fixed angles, one measurement is always enough, you never need to enter two values to get a full solution.

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Standard Formula Based on the exact 45-45-90 ratio: hypotenuse = leg × √2. No rounding in intermediate steps.

Known Value

What do you know?

Leg length (a) both legs are equal

units

45-45-90 Triangle

Both legs (a) are equal. Hypotenuse c = a√2.

45-45-90 Triangle Results

Leg (a)

Both legs equal

Hypotenuse (c)
-
c = a × √2

Area

a² / 2

Perimeter

2a + a√2

Working: Enter a value above to see the working.

All Measurements

Leg (a)-

Hypotenuse (c)-

Angle A45.000 deg

Angle B45.000 deg

Angle C (right angle)90.000 deg

Perimeter-

Area and Height

Area-

Height to hypotenuse (h)-

Inradius (r)-

Circumradius (R)-

Leg / hypotenuse ratio1 : √2 (1 : 1.4142)

Hypotenuse / leg ratio√2 : 1 (1.4142 : 1)

What is a 45-45-90 Triangle?

A 45-45-90 triangle is a special right triangle with two 45-degree angles and one 90-degree angle. Because both acute angles are equal, the triangle is also isosceles, meaning both legs (the two shorter sides) are identical in length. The hypotenuse (the side opposite the right angle) is always exactly the leg length multiplied by the square root of 2.

This triangle appears constantly in geometry, trigonometry, construction, and design. It is the shape you get when you cut a square diagonally in half. The 45-45-90 triangle is one of only two special right triangles with fixed angle ratios (the other being the 30-60-90 triangle).

The 45-45-90 Triangle Formulas

Measurement	Formula	Example (leg = 5)
Leg (a)	a (given, or derived below)	5
Hypotenuse (c)	c = a × √2	5 × 1.41421 = 7.07107
Area	Area = a² / 2	5² / 2 = 12.5
Perimeter	P = 2a + a√2 = a(2 + √2)	5 × 3.41421 = 17.07107
Height to hypotenuse (h)	h = a / √2 = c / 2	5 / 1.41421 = 3.53553
Inradius (r)	r = a(√2 - 1) / √2 = (a / √2) × (√2 - 1)	5 × 0.29289 = 1.46447
Circumradius (R)	R = c / 2 = a√2 / 2	7.07107 / 2 = 3.53553

Deriving the Formulas from Pythagoras

The 45-45-90 ratios follow directly from the Pythagorean theorem. If both legs are equal to a, then:

a² + a² = c²
2a² = c²
c = √(2a²) = a√2

This is why the hypotenuse is always exactly the leg times the square root of 2, regardless of the actual size of the triangle. The ratio is exact, not an approximation. When someone says "multiply by 1.41421", they are using a rounded decimal approximation of √2. The calculator above uses the full precision of Math.SQRT2 in JavaScript, which is accurate to 15 significant figures.

Worked Example Using the Default Values

With a leg length of 5 units:

Hypotenuse = 5 × √2 = 7.07107 units
Area = 5² / 2 = 25 / 2 = 12.5 square units
Perimeter = 5 + 5 + 7.07107 = 17.07107 units
Height to hypotenuse = 5 / √2 = 3.53553 units

Finding the Leg from the Hypotenuse

If you know the hypotenuse c and need the leg, rearrange c = a√2:

a = c / √2 = c × √2 / 2

For example, if the hypotenuse is 10: a = 10 / √2 = 10 × √2 / 2 = 5√2 = approximately 7.07107 units.

Finding the Leg from the Area

Area = a² / 2, so a² = 2 × Area, giving a = √(2 × Area).

For example, if the area is 50 square units: a = √(2 × 50) = √100 = 10 units.

Real-World Uses of the 45-45-90 Triangle

Roof framing: a 45-degree pitch means equal rise and run, forming a 45-45-90 triangle.
Tile cutting: diagonal cuts across square tiles create two 45-45-90 triangles.
Navigation and surveying: bearings at 45 degrees and equal north/east displacements.
Computer graphics: pixel-perfect diagonal lines on a grid at 45 degrees.
Woodworking: mitre joints at 45 degrees join two boards at a right angle.

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Method: All calculations use exact arithmetic with JavaScript's Math.SQRT2 constant (√2 = 1.41421356237...) and Math.sqrt(). No intermediate rounding is applied. Results are displayed rounded to 5 decimal places for display purposes.