Area of a Right Triangle Calculator
This calculator finds the area of a right triangle, plus its hypotenuse, perimeter and all three interior angles, from the two legs that meet at the right angle: the base and the height. Enter the base and height in the units of your choice, cm, m, mm, km, inches, feet or yards, and every result updates instantly. Area is worked out using the standard formula, half the base multiplied by the height, since a right triangle is exactly half of a rectangle with matching sides. The hypotenuse, the longest side, opposite the right angle, is calculated with the Pythagorean theorem: the square root of base squared plus height squared. From there you get the perimeter and both acute angles, worked out using arctan of the ratio between the two legs, with the right angle fixed at 90 degrees and all three angles summing to 180. A full breakdown, plus a worked example using the default values of base 6 and height 8, sits below the results so you can see how each figure was reached. This suits schoolwork, checking your own working, or practical jobs such as squaring up a wall, working out a roof pitch, or measuring rafter lengths. Make sure your base and height are entered in the same units before reading off the results, since the calculator does not convert between units.
1. Triangle Dimensions
2. Quick Reference
A right triangle has one angle of exactly 90°. The two shorter sides (legs) are called the base and height. The longest side, opposite the right angle, is the hypotenuse.
Formulae used:
Area = ½ × base × height
Hypotenuse = √(base² + height²)
Angle A = arctan(height ÷ base)
Angle B = arctan(base ÷ height)
Perimeter = base + height + hypotenuse
Full Measurements
All Angles
Worked Example (default values)
Given: base = 6 units, height = 8 units
Area = ½ × 6 × 8 = 24 units²
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10.00 units
Perimeter = 6 + 8 + 10 = 24.00 units
Angle A = arctan(8 ÷ 6) = arctan(1.3333) = 53.13°
Angle B = 90° − 53.13° = 36.87°
How to Calculate the Area of a Right Triangle
A right triangle (also called a right-angled triangle) is a triangle with one interior angle equal to exactly 90°. The two sides that form the right angle are called the legs (or base and height). The side opposite the right angle is the hypotenuse, which is always the longest side.
The area formula is straightforward: multiply the two legs together and divide by two. This works because a right triangle is exactly half of a rectangle with the same base and height.
| Measurement | Formula | Example (base=6, height=8) |
|---|---|---|
| Area | ½ × base × height | ½ × 6 × 8 = 24 units² |
| Hypotenuse | √(base² + height²) | √(36 + 64) = 10 units |
| Perimeter | base + height + hypotenuse | 6 + 8 + 10 = 24 units |
| Angle A (at base) | arctan(height ÷ base) | arctan(8÷6) = 53.13° |
| Angle B (at height) | arctan(base ÷ height) | arctan(6÷8) = 36.87° |
The Pythagorean Theorem
The Pythagorean theorem states that in any right triangle: a² + b² = c², where a and b are the legs and c is the hypotenuse. This is one of the most fundamental relationships in geometry. The 3-4-5 triangle is the simplest Pythagorean triple with integer sides: 3² + 4² = 9 + 16 = 25 = 5². The default example in this calculator (6, 8, 10) is a scaled version of the 3-4-5 triple.
Common Right Triangle Applications
Right triangles appear throughout everyday life and practical tasks:
- Building and construction: checking that walls and floors are square (the 3-4-5 rule), calculating roof pitch, and working out rafter lengths.
- Land surveying: measuring plots and calculating areas where survey lines meet at right angles.
- Navigation: calculating straight-line distance when you know north-south and east-west displacements.
- Screen and display sizing: diagonal measurement of screens from width and height.
- Physics and engineering: resolving forces, vector components, and inclined planes.
Special Right Triangles
Two right triangles have especially useful angle ratios:
- 45-45-90 triangle: both legs are equal, and the hypotenuse is leg × √2. For example, a triangle with legs of 5 has a hypotenuse of 5√2 ≈ 7.07.
- 30-60-90 triangle: the short leg is half the hypotenuse, and the long leg is short leg × √3. These occur when you cut an equilateral triangle in half.
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Method: Area = ½ × base × height (standard formula for a right triangle). Hypotenuse calculated using the Pythagorean theorem: c = √(a² + b²). Angles calculated using inverse tangent (arctan). All calculations are performed in the browser and are exact to floating-point precision.
Results are mathematically exact for the values entered. Ensure your base and height are in the same units before using this calculator.