Same shape, three sizes

2026-05-11

A right triangle has a secret: it is made of two smaller copies of itself. Drop a perpendicular from the right angle to the hypotenuse, and the two pieces left behind are scaled-down versions of the whole. That's all you need.

The altitude from the right-angle vertex to the hypotenuse cuts the big triangle into a blue piece and a red piece. Each has a right angle where the altitude lands; each shares one of the original triangle's acute angles. Same angle set, same shape - that is similarity.

Areas scale with the square

If you scale a shape up by a factor of k, every length gets longer by k, but the area gets bigger by k². A square of side 2 has four times the area of a square of side 1. A triangle twice as big has four times the area. A circle. A pentagon. Anything. This is the one fact the proof needs.

The big triangle's hypotenuse is c. The blue piece's hypotenuse is a. The red piece's hypotenuse is b. All three triangles are the same shape; what differs is the scale. Compared to the big one, the blue piece is scaled by a/c and so has area (a/c)² × big. The red piece is scaled by b/c and has area (b/c)² × big.

And of course - the blue and the red, taken together, are the big triangle. So:

(a/c)² + (b/c)² = 1

Multiply both sides by c² and the theorem walks out:

a² + b² = c²

Why this proof feels different

Bhāskara's proof and Garfield's proof are both rearrangement: two ways of measuring the same area, identity falls out. Euclid's proof is constructive: morph the leg-squares into rectangles. This one is neither. The whole theorem hinges on a single fact about any shape: scaling lengths by k scales areas by k². The right triangle is just where that fact gets to perform.

Einstein, who proved a version of this at age 11, said this is the proof he liked best. It is short, it uses no construction outside the triangle, and it leans on a fact that feels universal rather than geometric. The argument would survive in a world where squares and rectangles had been forgotten.

Same shape. Three sizes. The areas have to add up.

Other proofs of a² + b² = c²: Bhāskara's rearrangement, Garfield's trapezoid, Euclid's windmill.