Pythagorean Theorem

a^2 + b^2 = c^2 — the backbone of right-triangle geometry.

The idea

Theorem (Pythagorean theorem).

In a right triangle with legs $a$ and $b$ and hypotenuse $c$, the side opposite the right angle, $a^{2} + b^{2} = c^{2}.$

The hypotenuse is always the longest side. The theorem is a statement about areas: build a square outward on each of the three sides, and the square on the hypotenuse has exactly the combined area of the two squares on the legs.

It turns measurement into calculation. Two sides of a right triangle determine the third: $c = \sqrt{a^{2} + b^{2}}$ for the hypotenuse, and $\sqrt{c^{2} - a^{2}}$ for a missing leg. The converse holds as well — if three lengths satisfy $a^{2} + b^{2} = c^{2}$ with $c$ the longest, they form a right triangle, which is how a builder squares up a corner with a knotted rope.

A few whole-number triples are worth recognizing on sight: $3$-$4$-$5$, $5$-$12$-$13$, $8$-$15$-$17$.

Ways to work on it

• Walkthrough. Prove a^2 + b^2 = c^2 by dissection, then find a hypotenuse and a leg.
• Practice. Find a missing side — a hypotenuse or a leg.
• Hardest. Use the theorem to compute distances and its converse to classify triangles.

Not sure where to start? Take the ten-question placement test.