SOLVE RIGHT TRIANGLE GUIDE
How to find the hypotenuse
The hypotenuse is the side opposite a right triangle’s 90° angle. It is always longer than either leg. In Solve Right Triangle, it is labelled c; the perpendicular legs are a and b.
When both legs are known
Use the Pythagorean theorem. Square each leg, add the squares, and take the square root. Keep both measurements in the same unit before calculating.
c = √(a² + b²)
Example: legs of 6 cm and 8 cm
c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
The result passes a useful check: 10 cm is greater than both 6 cm and 8 cm. The area is (6 × 8) / 2 = 24 cm².
When one leg and an acute angle are known
First identify the leg relative to that angle. The opposite leg faces the angle; the adjacent leg touches it. The hypotenuse does not change when you switch angles, but opposite and adjacent do.
- Known opposite leg: c = opposite / sin(angle).
- Known adjacent leg: c = adjacent / cos(angle).
For an opposite leg of 5 m and an acute angle of 30°, c = 5 / sin(30°) = 5 / 0.5 = 10 m.
Avoid these mistakes
- Do not add the leg lengths directly. 6 + 8 is 14, but the hypotenuse is 10.
- Do not use this formula for a triangle without a 90° angle.
- Do not combine metres and centimetres before converting to a common unit.
- Keep full precision until the final answer.
Formula reference
For an independent explanation of these relationships, see Pearson’s right triangle formulas and worked examples.
Related right triangle guides
- How to find a missing right triangle angle
- SOH CAH TOA, explained visually
- 30–60–90 and 45–45–90 triangles
- Right triangles in practical measurements
- How to find a missing side of a right triangle
Open the right triangle calculator to check your own measurements.