SOLVE RIGHT TRIANGLE GUIDE
How to find a missing side of a right triangle
To find a missing side of a right triangle, you need either the other two sides or one side and one acute angle. The 90° angle is already known. Two angles alone do not establish a length: many triangles can share those angles at different sizes.
Missing hypotenuse: add the squared legs
The hypotenuse, c, faces the right angle. When legs a and b are known, use c = √(a² + b²). For legs of 9 cm and 12 cm, c = √(81 + 144) = √225 = 15 cm.
Missing leg: subtract from the squared hypotenuse
If you know c and b, rearrange a² + b² = c² to isolate a. The same method works with a known a and missing b.
b = √(c² − a²)
Example: c = 13 ft and b = 12 ft
a = √(13² − 12²) = √(169 − 144) = √25 = 5 ft.
Check: 5² + 12² = 169 = 13². The hypotenuse is longer than both legs.
One side and an angle: use trigonometry
Identify opposite and adjacent relative to the angle you know. In our diagram, a is opposite A and b is adjacent to A. Sine compares opposite to hypotenuse; cosine compares adjacent to hypotenuse; tangent compares opposite to adjacent.
- Known c: a = c × sin(A), b = c × cos(A).
- Known a: c = a / sin(A), b = a / tan(A).
- Known b: a = b × tan(A), c = b / cos(A).
For b = 4 m and A = 45°, a = 4 × tan(45°) = 4 m, and c = 4 / cos(45°) ≈ 5.657 m.
Before you calculate
Convert lengths to the same unit, verify that the triangle contains a right angle, and keep full precision until the final answer. A hypotenuse shorter than a known leg cannot form a right triangle. For a missing leg, subtract squared lengths before taking the square root; √c² − √b² gives a different result.
Solve the missing-leg exampleFormula reference
For an independent explanation of these relationships, see Pearson’s right triangle formulas and worked examples.
Related right triangle guides
- How to find the hypotenuse
- 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
Open the right triangle calculator to check your own measurements.