Pythagorean Theorem Calculator
The missing side of a right triangle, plus its angles and area.
Work out Pythagorean Theorem. The missing side of a right triangle, plus its angles and area. Every related measurement, not only the one asked for.
Hypotenuse (units)
5
3, 4, 5 — a Pythagorean triple
a² + b² = c². 3² + 4² = 25, and its square root is 5. All three sides are whole numbers, which makes this a Pythagorean triple — rare enough that the small ones have been known for four thousand years. The two acute angles always sum to 90°, since the third is the right angle.
How the Pythagorean Theorem Calculator works
a² + b² = c². This solves for the hypotenuse from two legs, or for a missing leg from the hypotenuse and one leg, and returns the area, perimeter and both acute angles alongside. It also flags when all three sides come out whole — a Pythagorean triple.
Also known as: hypotenuse calculator · a squared plus b squared · missing side of a right triangle · right triangle calculator
The theorem itself
a² + b² = c², where c is the side opposite the right angle. It holds for every right triangle and for no other kind, which is what makes it a test for squareness as well as a way to find a length.
That is why builders check a corner by measuring 3 along one wall, 4 along the other, and confirming the diagonal is exactly 5. If it is not, the corner is not square.
In practice
Legs of 3 and 4 give 9 + 16 = 25, and the hypotenuse is 5. The area is 6 and the two acute angles are 36.87° and 53.13°, always summing to 90 because the third angle takes the remaining 90.
Working backwards, a hypotenuse of 5 with a leg of 3 gives √(25 − 9) = 4.
A leg can never equal or exceed the hypotenuse. The hypotenuse faces the largest angle, so it is always the longest side — and this page rejects that combination rather than returning an imaginary number.
Pythagorean triples
When all three sides come out whole, the triangle is a Pythagorean triple. 3-4-5 and 5-12-13 are the smallest, and both were known to Babylonian mathematicians nearly four thousand years ago.
They are rare. Most right triangles have an irrational hypotenuse — a unit square's diagonal is √2, which was reportedly a scandal when the Pythagoreans discovered no fraction equals it.
Frequently asked questions
How do I find the hypotenuse?
Square both legs, add them, take the square root. A 3 and a 4 give √25 = 5 — the most famous right triangle there is.
How do I find a missing leg?
Subtract the known leg squared from the hypotenuse squared, then take the square root. A hypotenuse of 5 with a leg of 3 gives √16 = 4.
What is a Pythagorean triple?
A right triangle whose three sides are all whole numbers, such as 3-4-5 or 5-12-13. They are rare enough that the small ones have been known for four thousand years.
Why can't a leg be longer than the hypotenuse?
Because the hypotenuse is opposite the right angle, which is the largest angle in the triangle, so it always faces the longest side. This page rejects that combination rather than returning an imaginary number.
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