Pythagorean Theorem Calculator

The Most Famous Equation in Geometry

The Pythagorean theorem is the rule that, in a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides: a² + b² = c². It is probably the most recognisable equation in all of mathematics, and for good reason — it is the bridge between the lengths of a triangle and the right angle inside it, and from it flows the entire idea of distance. Knowing any two sides of a right triangle, you can always find the third: to find the hypotenuse, square the two legs, add them, and take the square root; to find a missing leg, subtract the known leg's square from the hypotenuse's square and take the root. This calculator does whichever you need automatically — just leave the unknown side blank — and shows the formula filled in so the method is clear, not hidden. It also enforces the one rule beginners forget: the hypotenuse is always the longest side, so it cannot be shorter than a leg.

What makes the theorem so useful is that "find the third side of a right triangle" is a question that hides inside an enormous range of practical problems. The straight-line distance between two points on a map or screen is a hypotenuse — which is exactly why the distance formula is just the Pythagorean theorem in disguise. Construction and carpentry rely on it constantly: the "3-4-5 rule" for squaring a corner is the theorem in its most famous whole-number form, since 3² + 4² = 5². Navigation, computer graphics, physics (resolving vectors into components), and even the diagonal of your television screen all come back to a² + b² = c². The theorem extends cleanly into three dimensions too: the longest diagonal through a box, corner to opposite corner, is found by applying the theorem twice, which the 3D mode here does for you from the length, width, and height.

Older Than Pythagoras

Although it carries Pythagoras's name from around 500 BC, the relationship was known and used much earlier — Babylonian clay tablets list Pythagorean triples (whole-number right triangles) more than a thousand years before him, and builders in Egypt and India used the 3-4-5 triangle to lay out right angles. What the Greeks added was proof: a demonstration that the relationship holds for every right triangle, not just the convenient whole-number ones, and that is the leap from a useful trick to a theorem. There are now hundreds of distinct proofs, including one by a future US president. For everyday work, though, the value is immediate and practical: whenever a right angle and two lengths meet, this calculator gives the third length instantly and privately in your browser, with the working shown — for homework, a building project, a design, or simple curiosity.