How to Solve a Right Triangle Step by Step

Right-triangle trigonometry · 6 minute read

"Solving" a triangle means finding all three sides and both acute angles. You always know the right angle (90°), so you need two more pieces of information, at least one of them a side.

A reliable method

Sketch the triangle and label the known values.
If you know an angle, the other acute angle is 90° − that angle.
Choose the trig ratio that links a known value and the unknown side.
Solve, then check with the Pythagorean theorem.

Example: one angle and one side

Angle A = 35° and the hypotenuse c = 20. Find the other angle and both legs.

Angle B = 90° − 35° = 55°
Side opposite A: a = 20 · sin 35° ≈ 11.47
Side adjacent to A: b = 20 · cos 35° ≈ 16.38

Check: 11.47² + 16.38² ≈ 131.6 + 268.3 ≈ 399.9 ≈ 20². ✓

Example: two sides

The legs are a = 7 and b = 24. Find c and both angles.

c = √(7² + 24²) = √625 = 25
A = tan⁻¹(7/24) ≈ 16.3°
B = 90° − 16.3° ≈ 73.7°

Tips

Prefer the given numbers over calculated ones, so rounding errors do not build up.
Check that the calculator is in degree mode.
The largest angle is opposite the longest side.

Practice it

Practice solving right triangles