Pythagorean Theorem Calculator
Use the Pythagorean theorem to find a missing side of a right triangle, with the formula, a worked example and common triples.
The most useful theorem in geometry
The Pythagorean theorem relates the three sides of a right-angled triangle, and it is probably the single most useful result in practical geometry. Given any two sides of a right triangle, it finds the third. This applies it to find a missing side from the two you know.
The theorem
For a right triangle, the square of the longest side (the hypotenuse, opposite the right angle) equals the sum of the squares of the other two sides:
a squared + b squared = c squaredc is the hypotenuse (the longest side)To find the hypotenuse: c = square root of (a squared + b squared)To find a shorter side: a = square root of (c squared - b squared)Worked example and common triples
A classic example uses the simplest whole-number right triangle:
Certain whole-number side combinations, called Pythagorean triples, crop up often and are worth recognising:
| Triple | Check |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 |
| 5, 12, 13 | 25 + 144 = 169 |
| 8, 15, 17 | 64 + 225 = 289 |
| 7, 24, 25 | 49 + 576 = 625 |
The theorem is everywhere in practice: checking a corner is square in construction (the 3-4-5 method), finding the straight-line distance between two points, working out a ramp or roof length. It is also the basis of the distance formula in coordinate geometry, which is really Pythagoras applied to the gap between two points.
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 other two sides — a squared plus b squared equals c squared.
How do I find a missing side?
For the hypotenuse, take the square root of the sum of the other two sides squared. For a shorter side, subtract before taking the root.
What is a Pythagorean triple?
A set of whole numbers that form a right triangle, like 3-4-5 or 5-12-13 — worth recognising because they come up often.
Where is it used in real life?
Checking corners are square in construction, finding straight-line distances, and working out ramp or roof lengths, among much else.
Is anything uploaded?
No. The calculation happens in your browser.