Why a² + b² = c²

2026-05-01

Take any right triangle. Make four copies. Slide them around inside a slightly bigger square. Watch what stays the same.

The pale yellow region is whatever is left over after the four triangles take their share. On the left, that leftover is a single tilted square. Its side is the hypotenuse, so its area is c². Slide all the way to the right and the leftover splits into two upright squares: one of side a, one of side b. Its area is a² + b².

The four triangles never disappear. They cover the same total area at every point in the slide. So the yellow leftover is the same area at every point. The two configurations are the same area. That is the theorem.

Try other shapes

Drag the blue and red dots in the top panel to reshape the triangle. The classic 3-4-5 is the default. Try setting a = b and watch the two upright squares come out identical. Try a tall, skinny one (a ≈ 1, b ≈ 6) and watch the leftover region elongate. Try anything you like.

The proof works for every shape. There is nothing special about 3-4-5; that triangle just happens to have whole-number sides, which is why the Greeks and Egyptians built rope-stretching crews around it. Any right triangle in the universe satisfies the same identity.

Why this is one of the better proofs

There are dozens of proofs of Pythagoras' theorem. This one - usually attributed to the 12th-century Indian mathematician Bhāskara II, though the diagram is older - is loved because it asks you to do nothing except look. No equations, no similar-triangle ratios, no algebra. Just the same four pieces, the same total empty space, two ways of putting them together.

Bhāskara is said to have presented the diagram with the single word "Behold!". We will not be doing that.

c² = a² + b². No tricks. No new idea. The same shape, seen twice.