Pythagorean Theorem

Geometry · Axiom Academy

One of the most famous and useful theorems in all of mathematics — the rule that ties the three sides of every right triangle together. Stand a square on each side of a right triangle. The area of the square on the hypotenuse (the longest side, opposite the right angle) is exactly the sum of the areas of the squares on the other two sides — the legs . For a 3–4–5 triangle that reads 9 + 16 = 25. Because the three sides are locked together, knowing any two of them pins down the third. Substitute the two you know into a² + b² = c² and solve. Watch it play out on the classic 3–4–5 triangle: the two legs determine the hypotenuse. Know both legs? Square them, add, take the square root: c = √(a² + b²). Know the hypotenuse and one leg? Subtract, then square-root: a = √(c² − b²). Hypotenuse, 3 and 4: 3² + 4² = 9 + 16 = 25, so c = √25 = 5. A leg, given hypotenuse 13 and leg 12: a² + 12² = 13² → a² = 169 − 144 = 25, so a = 5. A non-integer answer, legs 5 and 7: 5² + 7² = 25 + 49 = 74, so c = √74 ≈ 8.60. Not every triangle gives a whole number. Pythagorean triples are sets of three positive integers that satisfy the theorem exactly. Spotting one saves you the square-root work. A few are worth memorizing — and any whole-number multiple of a triple is itself a triple, so scaling a 3–4–5 up to 6–8–10 keeps it a right triangle. Three quick cases: a whole-number hypotenuse, a whole-number leg, and a square-root answer.

This is the written version of the interactive lesson above. See the full Geometry course.