Pythagorean Theorem Calculator
Enter any two sides; leave the third blank.
- Using a² + b² = c²
- c = √(3² + 4²) = √(9 + 16) = √25
- c = 5
a² + b² = c²
The Pythagorean Theorem
The Pythagorean theorem states that in a right triangle with legs a and b and hypotenuse c, the relationship a² + b² = c² holds. It is one of the most famous results in mathematics, with hundreds of proofs spanning geometry, algebra, and even physics.
Given two legs, find the hypotenuse: c = √(a² + b²). Given one leg and the hypotenuse, find the other leg: b = √(c² − a²). The calculator shows each substitution and arithmetic step so you can follow the algebra.
Classic examples: the 3-4-5 triangle gives 3² + 4² = 9 + 16 = 25 = 5². The 5-12-13 triangle gives 25 + 144 = 169 = 13². Any multiple of a Pythagorean triple is also valid — (6, 8, 10) is twice (3, 4, 5).
The hypotenuse is always the longest side, opposite the right angle. If your computed hypotenuse would be shorter than a leg, the inputs cannot form a right triangle and the calculator returns an error.
An SVG diagram labels sides a, b, and c with the right angle marked. Leave exactly one field blank, enter the other two, and the missing side appears with full working shown below.
Examples
| Example | Result |
|---|---|
| a=3, b=4 | c = 5 |
| a=5, c=13 | b = 12 |
| b=8, c=10 | a = 6 |
| Verify 3²+4² | 25 |
Frequently asked questions
c is the hypotenuse — the longest side, opposite the right angle.
No. The Pythagorean theorem applies only to right triangles.
Yes. Leave the unknown leg blank and enter c and the other leg.