Pythagorean Theorem Calculator
Find the hypotenuse of a right triangle from its two legs — plus area and perimeter.
- Hypotenuse (c)5.0000
- Triangle area6.0000
- Perimeter12.0000
Inputs used
- Leg a3
- Leg b4
About this calculator
The Pythagorean theorem says that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: c² = a² + b². Enter the legs and the hypotenuse is the square root of that sum — the classic 3-4-5 triangle gives √(9 + 16) = 5.
To find a missing leg instead, rearrange: a = √(c² − b²). Enter the hypotenuse as one "leg" here and take the result with the larger value as c — or simply check that the numbers satisfy a² + b² = c².
The theorem only holds for right triangles, but it hides everywhere: diagonal of a TV screen, the shortest path across a rectangular field, distance between two points on a map.
Quick reference
Hypotenuse by leg lengths
The hypotenuse c = √(a² + b²) for each pair of legs a (rows) and b (columns).
| Leg a | 4 | 6 | 8 | 10 | 12 | 16 |
|---|---|---|---|---|---|---|
| 3 | 5.0000 | 6.7082 | 8.5440 | 10.4403 | 12.3693 | 16.2788 |
| 5 | 6.4031 | 7.8102 | 9.4340 | 11.1803 | 13.0000 | 16.7631 |
| 6 | 7.2111 | 8.4853 | 10.0000 | 11.6619 | 13.4164 | 17.0880 |
| 8 | 8.9443 | 10.0000 | 11.3137 | 12.8062 | 14.4222 | 17.8885 |
| 9 | 9.8489 | 10.8167 | 12.0416 | 13.4536 | 15.0000 | 18.3576 |
| 12 | 12.6491 | 13.4164 | 14.4222 | 15.6205 | 16.9706 | 20.0000 |
| 15 | 15.5242 | 16.1555 | 17.0000 | 18.0278 | 19.2094 | 21.9317 |
| 20 | 20.3961 | 20.8806 | 21.5407 | 22.3607 | 23.3238 | 25.6125 |
Frequently asked questions
What is the Pythagorean theorem?
For a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Rearranged, the hypotenuse is c = √(a² + b²), so legs of 3 and 4 give √(9 + 16) = √25 = 5.
How do I find a missing leg instead of the hypotenuse?
Subtract instead of add: a = √(c² − b²), where c is the hypotenuse. For a hypotenuse of 13 and one leg of 5, the other leg is √(169 − 25) = √144 = 12.
What are the most common Pythagorean triples?
Whole-number right triangles like 3-4-5, 5-12-13, 8-15-17 and 7-24-25 — sets where a² + b² = c² comes out exact. Any multiple works too, so 6-8-10 is just the 3-4-5 triangle doubled.
Does the Pythagorean theorem work for any triangle?
No — only right triangles, where one angle is exactly 90°. For other triangles you need the law of cosines, which adds a correction term and reduces to a² + b² = c² only when the angle between the legs is a right angle.
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