Pythagorean Theorem Calculator

The hypotenuse is always the odd one out

a² + b² = c² holds only when c is the hypotenuse, the side opposite the right angle and always the longest. Solving for a leg rearranges to a² = c² − b², a subtraction rather than an addition. Adding when you should subtract is the dominant error here, and it produces an answer larger than the hypotenuse, which is the check that catches it immediately.

It only applies to right triangles

For any other triangle the relationship becomes the law of cosines, c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras exactly when C is 90° and the cosine term vanishes. Applying Pythagoras to a triangle that is not right-angled gives a plausible number and a wrong one, with nothing in the arithmetic to signal the problem.

Pythagorean triples are worth recognising

3-4-5, 5-12-13, 8-15-17 and 7-24-25 are integer-sided right triangles, and their multiples work too — 6-8-10, 9-12-15. Recognising them saves calculation, and the 3-4-5 triple has a practical use that predates the theorem's proof: measuring 3 units along one edge and 4 along another, then adjusting until the diagonal is exactly 5, squares a corner with nothing but a tape measure. Builders still use it.

Distance between two points is the same theorem

The distance formula √((x₂−x₁)² + (y₂−y₁)²) is Pythagoras applied to the horizontal and vertical differences. It extends to three dimensions by adding the z difference under the same root, which is how diagonal measurements inside a box, screen sizes and 3D coordinate distances are all calculated.

Where it shows up in practice

Screen and TV sizes are the diagonal, which is why a 27-inch 16:9 monitor is about 23.5 inches wide and 13.2 tall — comparing diagonals across different aspect ratios misleads badly. Checking a room or a frame is square, sizing a cable run diagonally across a ceiling, and rafter lengths in roofing all reduce to the same calculation.

Watch the units and the rounding

All three sides must be in the same unit before squaring; mixing feet and inches produces nonsense that looks like a valid number. And square roots are usually irrational, so a hypotenuse of √50 is 7.07 only after rounding. For anything being cut or built, keep the exact value through intermediate steps and round once at the end.