The formula: a² + b² = c²
For any right triangle, squaring each of the two shorter sides (the legs) and adding them together gives the square of the longest side (the hypotenuse, opposite the right angle). Taking the square root of that sum gives the hypotenuse itself: c = √(a² + b²).
The classic example: legs 3 and 4
c = √(3² + 4²) = √(9 + 16) = √25 = 5. The perimeter is the sum of all three sides: 3 + 4 + 5 = 12. This 3-4-5 triangle is the best-known example because every value works out to a whole number.
A second whole-number example, legs 5 and 12
c = √(5² + 12²) = √(25 + 144) = √169 = 13, giving a perimeter of 5 + 12 + 13 = 30. Sets of three whole numbers that satisfy a² + b² = c² exactly, like 3-4-5 and 5-12-13, are called Pythagorean triples.
What the theorem doesn't tell you
The Pythagorean theorem only ever produces the hypotenuse and perimeter — it says nothing about the triangle's area or its interior angles. Finding those requires the additional steps covered in the companion article on solving a full right triangle.