Pythagorean Theorem Calculator — hypotenuse from two legs

How it works

The Pythagorean theorem is the fundamental relationship among the sides of a right triangle — one with a 90° angle. If the two shorter sides (the legs) are a and b, and the longest side (the hypotenuse, opposite the right angle) is c, then:

a² + b² = c²

To find the hypotenuse, square both legs, add them, and take the square root:

c = √(a² + b²)

The same equation finds a missing leg if you know the hypotenuse: rearrange to a = √(c² − b²). It applies only to right triangles — the 90° angle is what makes the squares of the sides add up this exactly.

Worked example

A right triangle has legs 3 and 4. Square them, add, and take the root:

c = √(3² + 4²) = √(9 + 16) = √25 = 5

That's the famous 3-4-5 triangle — three whole numbers that fit exactly. Another is 5, 12, 13: √(25 + 144) = √169 = 13. Not every pair comes out whole, though — legs of 1 and 1 give c = √2 ≈ 1.414, an irrational number.

Frequently asked questions

How do I find the hypotenuse?

Square each leg, add the two squares, and take the square root of the total. For legs 6 and 8: √(36 + 64) = √100 = 10. The hypotenuse is always the longest side and sits opposite the right angle.

How do I find a missing leg instead?

Rearrange the theorem: if you know the hypotenuse c and one leg b, the other leg is a = √(c² − b²). For a hypotenuse of 13 and a leg of 5: √(169 − 25) = √144 = 12. Just make sure you subtract, since you're solving for a leg, not the hypotenuse.

Does it work for any triangle?

No — only **right triangles**, those with a 90° angle. For other triangles the relationship between the sides is different (the law of cosines generalises it). If a triangle isn't right-angled, a² + b² won't equal c².

What is a Pythagorean triple?

A set of three whole numbers a, b, c that satisfy a² + b² = c² exactly, like (3, 4, 5), (5, 12, 13) and (8, 15, 17). Any multiple of a triple is also one — (6, 8, 10) is just (3, 4, 5) doubled. Most side lengths don't form triples, giving an irrational hypotenuse instead.

Can I check whether a triangle is right-angled?

Yes — if the three side lengths satisfy a² + b² = c² (with c the longest), the triangle has a right angle; this is the converse of the theorem. For example 7, 24, 25 works (49 + 576 = 625), so it's a right triangle.

What units does the answer use?

Whatever units the legs are in — the theorem is unit-agnostic, so if a and b are in metres the hypotenuse is in metres too. This calculator treats the legs as plain numbers, so just keep both legs in the same unit and the result follows.