How this calculator works
In a right triangle, the square on the hypotenuse (the side opposite the right angle) equals the sum of the squares on the other two sides: a² + b² = c². To find the hypotenuse, add the squares of the legs and take the square root. To find a leg, subtract the square of the other leg from the square of the hypotenuse and take the square root.
Builders use the same rule to check a corner is square: measure 3 ft along one side and 4 ft along the other — the diagonal should be exactly 5 ft (or 6-8-10, 9-12-15 for bigger layouts).
Assumptions and tips
- The triangle must have a 90° angle; for other triangles use the law of cosines.
- Use the same unit for every side.
Worked example
Using the default values in the calculator above:
| Input | Value |
|---|---|
| Find | Hypotenuse c (know a and b) |
| Leg a | 3 |
| Leg b | 4 |
| Result | Value |
|---|---|
| Hypotenuse c | 5 |
| Angle A (opposite a) | 36.8699 ° |
| Angle B (opposite b) | 53.1301 ° |
| Area | 6 sq units |
| Perimeter | 12 units |
| Sides | a = 3, b = 4, c = 5 |
Step by step:
- c² = a² + b² = 9 + 16 = 25
- c = √25 = 5
Pythagorean triples (whole-number right triangles)
Useful for squaring corners on the job — any multiple works too (6-8-10, 9-12-15).
| Legs a, b | Hypotenuse c | Angle A |
|---|---|---|
| 3, 4 | 5 | 36.8699 ° |
| 5, 12 | 13 | 22.6199 ° |
| 8, 15 | 17 | 28.0725 ° |
| 7, 24 | 25 | 16.2602 ° |
| 20, 21 | 29 | 43.6028 ° |
| 9, 40 | 41 | 12.6804 ° |
| 6, 8 | 10 | 36.8699 ° |
| 12, 16 | 20 | 36.8699 ° |
| 1, 1 | 1.414214 | 45 ° |
Common mistakes to avoid
- Using the formula on a triangle without a right angle.
- Adding the sides before squaring: (3 + 4)² is not 3² + 4².
- Treating the longest side as a leg when solving for a missing side.
Frequently asked questions
What is the hypotenuse of a 3-4-5 triangle?
5: √(3² + 4²) = √25 = 5.
How do I find a missing leg?
Subtract: b = √(c² − a²). For c = 13 and a = 5, b = √(169 − 25) = 12.
How do I check if a corner is square?
Use the 3-4-5 rule: mark 3 and 4 units along the two sides; if the diagonal is 5 units, the corner is 90°.
How long is the diagonal of a 10 × 12 room?
√(100 + 144) = 15.62 ft.
Sources
Built and reviewed by Anirban Kundu, Creator & Editor · M.Sc Physics, 20 years teaching science