Five pieces. No equation.

2026-05-11

Take the larger leg-square. Mark its centre. Draw two lines through that centre - one parallel to the hypotenuse, one perpendicular. You now have four pieces. Add the smaller leg-square as a fifth. Slide them into the hypotenuse-square. They fit exactly.

That is the proof. No algebra, no rearrangement-with-leftover-areas, no "computed two ways". You watch five pieces - four pinwheel quadrilaterals and one whole square - flow into a single bigger square. The bigger square's area is c². The five pieces are worth a² + b². They occupy the same space. They are the same area.

Where the cuts come from

The two cuts through the centre of b² are not arbitrary. They are precisely parallel and perpendicular to the hypotenuse. This is what makes the four pieces fit around the central a² inside c². Because c²'s sides are themselves perpendicular to the hypotenuse, the pinwheel edges of the four pieces align with the inner edges of c² and the outer edges of a² exactly.

No piece is rotated. No piece is flipped. Every piece simply translates to its target. This is the most magical thing about Perigal's construction: rigid sliding, nothing else.

Henry Perigal, amateur

Henry Perigal was a London stockbroker who liked geometry. He published this dissection in 1873, age 72. He was so fond of it that he had the diagram engraved on his tombstone. It is one of the shortest and visually cleanest proofs of Pythagoras ever found, and it was discovered by a man who never made a living from mathematics.

The same dissection appears in the work of Thābit ibn Qurra in 9th-century Baghdad, eight centuries before Perigal. He did it first. But the construction was lost to Europe and rediscovered from scratch.

Five pieces. Two arrangements. Same area. That's the whole proof.

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