Pythagorean Theorem Calculator

Solve the hypotenuse or a missing leg of a right triangle.

Hypotenuse (c)

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Sides (a, b, c)

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Area

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Perimeter

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Acute angles

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Applies to right-angled triangles only. The hypotenuse must be longer than either leg.

The most famous equation in geometry

For any right-angled triangle, the area of the square on the hypotenuse equals the combined area of the squares on the other two sides. In symbols, that is the relationship below.

a² + b² = c²　⟹　c = √(a² + b²)

Knowing any two sides gives the third. To find the hypotenuse, add the squares of the legs and take the root; to find a leg, subtract the known leg’s square from the hypotenuse’s and take the root.

Worked example

With legs of 3 and 4, the hypotenuse is √(3² + 4²) = √25 = 5 — the classic 3-4-5 triangle. Its area is 6 and the acute angles are about 36.87° and 53.13°.

More useful than it looks

Beyond homework, the theorem squares up rooms and decking, finds diagonal screen sizes, sets out right angles on site, and underpins distance calculations in navigation and graphics. The humble 3-4-5 triangle is a builder’s best friend.

Handy to know

Right angles only. The relationship fails for other triangles.
Hypotenuse is longest. If your "leg" exceeds the hypotenuse, something is off.
Triples speed checks. 3-4-5, 5-12-13 and 8-15-17 are worth memorising.

Frequently asked questions

What does the theorem state?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It only holds for right-angled triangles.

How do I find a leg instead of the hypotenuse?

Rearrange to b = √(c² − a²). The hypotenuse must be the longest side, so it has to be larger than the known leg or there is no solution.

What is a Pythagorean triple?

A set of three whole numbers that fit the theorem, like 3-4-5, 5-12-13 or 8-15-17. They are handy for quick checks and for squaring up real-world layouts.

Can I use it to check a right angle?

Yes. Builders use the 3-4-5 method: measure 3 and 4 units along two sides, and if the diagonal is exactly 5, the corner is square.