The formula everyone memorizes
For any right triangle — one with a 90-degree angle — the theorem states:
a² + b² = c²
where a and b are the lengths of the two legs (the sides forming the right angle) and c is the hypotenuse (the side opposite the right angle, always the longest side). Most people learn this as a formula to plug numbers into. Fewer people see why it's actually true, which is a shame, because the reason is short and makes the formula much harder to misapply afterward.
A visual proof
Take a right triangle with legs a and b and hypotenuse c. Draw a square with side length a + b, and arrange four copies of the original triangle inside it, each rotated 90 degrees from the last, so their hypotenuses form a smaller, tilted square in the middle.
The big square's area is (a + b)². That same area also equals the four triangles' combined area, plus the tilted inner square's area: 4 × (½ab) + c², which simplifies to 2ab + c².
Setting the two expressions for the same area equal:
(a + b)² = 2ab + c²
a² + 2ab + b² = 2ab + c²
a² + b² = c²
The 2ab terms cancel on both sides, leaving exactly the theorem. This is one of dozens of known proofs (some sources count several hundred), but this particular one is worth walking through once because it shows the theorem isn't an arbitrary rule about right triangles — it falls directly out of comparing two ways of measuring the same square's area.
The mistake that shows up constantly
c is specifically the hypotenuse — the side opposite the right angle — not just "whichever side is labeled c" or "the longest side by assumption." Applying the formula with the wrong side substituted for c is the single most common error, especially in a triangle that isn't drawn with the right angle conveniently at the bottom. Before applying the formula, identify the right angle first, then identify which side sits opposite it — that side is c, regardless of how the triangle happens to be oriented on the page.
Using it to find a missing side
Given any two sides of a right triangle, the third follows directly:
- Missing hypotenuse:
c = √(a² + b²). If the legs are 3 and 4,c = √(9 + 16) = √25 = 5. - Missing leg: rearrange to
a = √(c² - b²). If the hypotenuse is 13 and one leg is 5, the other leg is√(169 - 25) = √144 = 12.
The 3-4-5 and 5-12-13 combinations above are the two most common "Pythagorean triples" — sets of whole numbers that satisfy the theorem exactly — and recognizing them on sight saves time on problems that use them, though the formula works identically for any right triangle, whole-number sides or not.
The converse: using it to test for a right angle
The theorem also runs in reverse: if a triangle's three side lengths satisfy a² + b² = c² (with c as the longest side), the triangle is guaranteed to have a right angle opposite that longest side — even without measuring any angle directly. This converse is genuinely useful outside of pure geometry problems: it's the basis for the "3-4-5 method" carpenters and builders use to confirm a corner is truly square, by measuring 3 units along one edge, 4 along the adjacent edge, and checking that the diagonal between those two points measures exactly 5.
Why the underlying idea outlasts the formula
Once the relationship between the areas is understood rather than just the equation, the theorem stops being a rule to look up and becomes something that can be reconstructed from scratch if forgotten — draw the square, split it into the triangles and the tilted inner square, compare the two ways of expressing the same total area. That's a more durable form of knowing the theorem than memorizing a² + b² = c² in isolation, and it's what makes the converse and the various extensions (like the distance formula in coordinate geometry, which is the Pythagorean theorem applied to horizontal and vertical distances) feel like natural consequences rather than separate facts to memorize on top of the original.