How the Pythagorean theorem is used
In a right triangle the square on the longest side equals the two other squares added together, so any two sides give the third.
a² + b² = c² · c = √(a² + b²) · a = √(c² − b²)
- Hypotenuse: square both legs, add them, then take the square root.
- Missing leg: square the hypotenuse, subtract the other leg squared, then take the square root.
- Check: if a² + b² equals c², the triangle is right. Bigger means acute, smaller means obtuse.
- The acute angles come from tan⁻¹(a ÷ b) and tan⁻¹(b ÷ a), and they always add to 90°.
Worked example — Legs 3 and 4: c = √(9 + 16) = √25 = 5. Area = ½ × 3 × 4 = 6, perimeter 12, angles 36.87° and 53.13°.
Common right triangles
| Leg a | Leg b | Hypotenuse | Acute angles |
|---|---|---|---|
| 3 cm | 4 cm | 5 cm | 36.87° and 53.13° |
| 5 cm | 12 cm | 13 cm | 22.62° and 67.38° |
| 8 cm | 15 cm | 17 cm | 28.07° and 61.93° |
| 1 cm | 1 cm | 1.4142 cm (√2) | 45° and 45° |
| 6 cm | 8 cm | 10 cm | 36.87° and 53.13° |
| 2 cm | 3 cm | 3.6056 cm (√13) | 33.69° and 56.31° |
Swipe the table sideways to see the rest.
Hypotenuse from two legs, in centimetres. Each row is worked out by this calculator.
Things to consider
- The hypotenuse is always opposite the right angle and is the longest side.
- Keep √ in the answer for exact work, and round only at the end.
- Multiply a triple by any whole number and you get another, like 6-8-10.
- Use the same unit for every side before you start.
Frequently asked questions
What is the Pythagorean theorem?
In a right triangle, a² + b² = c², where c is the hypotenuse, the side opposite the right angle.
How do I find the hypotenuse?
Square both legs, add them and take the square root. Legs 3 and 4 give √25 = 5. For equal legs, the 45-45-90 Triangle Calculator gives the result directly.
How do I find a missing leg?
Subtract the known leg squared from the hypotenuse squared, then take the square root. c 13 and b 5 give √144 = 12.
What are Pythagorean triples?
Whole-number sides that fit a² + b² = c², such as 3-4-5, 5-12-13 and 8-15-17.
How do I check a corner is square on a building site?
Measure 3 units along one side and 4 along the other. If the diagonal between the marks is 5, the corner is square. Scale up to 6-8-10 for bigger slabs.
How does this relate to the distance formula?
The distance between two points is the hypotenuse of a triangle with sides Δx and Δy. The Distance Calculator: Distance Between Two Points does this for coordinates.
Results are estimates for general information only, not financial, medical or professional advice.