How the right triangle is solved
Label the legs a and b, the hypotenuse c (the longest side, across from the right angle C), and the acute angles A and B, each across from the side with the same letter. Two facts do all the work:
sin A = a/c cos A = b/c tan A = a/b
A + B = 90°
With two sides, the Pythagorean theorem gives the third, and an inverse trig function gives an angle. With a side and an angle, sine, cosine or tangent gives the other sides. Two angles alone aren't enough: every triangle with the same angles has the same shape but could be any size.
The calculator also reports:
- Area = ½ab, because the legs are a base and its height.
- Altitude to the hypotenuse h = ab/c, from writing the area two ways: ½ab = ½ch.
- Hypotenuse segments a²/c and b²/c, the two pieces the altitude cuts the hypotenuse into.
- Inradius (a + b − c)/2 and circumradius c/2. The hypotenuse is always a diameter of the circumscribed circle.
Worked example
A ladder 10 ft long leans against a wall at 30° from the ground. How high does it reach, and how far is its foot from the wall?
- The ladder is the hypotenuse, c = 10. Put angle A = 60° at the top, so the angle at the ground is B = 30°.
- Height up the wall (side b, across from B): b = c·sin B = 10 × 0.5 = 5 ft.
- Distance from the wall (side a): a = c·cos B = 10 × 0.866025 = 8.660254 ft.
- Area = ½ × 5 × 8.660254 = 21.650635 sq ft, and the perimeter is 23.660254 ft.
Enter c = 10 and B = 30 above to see the same numbers, or the classic 3-4-5 triangle that loads by default.
Common right triangles
| Triangle | Sides | Angles |
|---|---|---|
| 3-4-5 | 3, 4, 5 | 36.87°, 53.13°, 90° |
| 5-12-13 | 5, 12, 13 | 22.62°, 67.38°, 90° |
| 8-15-17 | 8, 15, 17 | 28.07°, 61.93°, 90° |
| 45-45-90 | 1, 1, √2 | 45°, 45°, 90° |
| 30-60-90 | 1, √3, 2 | 30°, 60°, 90° |
For exact answers with radicals on the two special triangles, use the 30-60-90 and 45-45-90 calculator.
When to use a different tool
This calculator assumes one angle is exactly 90°. For any other triangle, including ones solved with the law of sines or cosines (SSS, SAS, ASA, AAS or SSA), use the triangle calculator. If you only need the missing side, the Pythagorean theorem calculator is quicker. Measurements in real life carry error, so a triangle built from measured sides may be slightly off from 90°.