Pythagorean theorem

The Pythagorean theorem says that in any right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is the side opposite the right angle and it is always the longest side, so it is the one that gets squared alone on its own side of the equation.

Using it has three steps. Name the two known sides and the unknown one, check which side is the hypotenuse, substitute, then take the square root to get back to a length. For legs of 3 and 4 the theorem gives 9 + 16 = 25, so the hypotenuse is 5 — the 3-4-5 triangle, one of the common triples worth recognizing along with 5-12-13 and 6-8-10, which are its multiples.

Two mistakes account for most of the misses. The first is squaring the wrong side: the hypotenuse is the one squared alone, so when the missing side is a leg the equation is rearranged to c² − a² = b² and the answer is still a square root. The second is using it at all on a triangle with no right angle — the theorem names the right angle for a reason, and a triangle without one does not obey the rule.