Pythagorean Theorem · PythagoLab

Puzzle

Each side of a right triangle carries a square. The Pythagorean theorem says the two smaller squares together have the same area as the square on the hypotenuse: a² + b² = c².

Don't calculate yet; try it: b² is already cut into 4 pieces. Drag them and a² into c² and see if they fit.

Out-in complement

In ancient China the legs of a right triangle were called gou and gu and the hypotenuse xian. Zhao Shuang proved the theorem with his diagram, and Liu Hui by “out-in complement”.

Out-in complement: move the parts you cut off elsewhere; the area does not change.

Euclid's proof

Proposition 47 of Book I of the Elements. A perpendicular from the right angle C splits c² into two rectangles, which are shown to equal a² and b².

The animation replaces Euclid's congruent triangles with three area-preserving moves: shear, rotate, shear.

More proofs

The theorem may have more proofs than any other; collections list over three hundred. Here are two more: the Pythagorean rearrangement and US President Garfield's trapezoid.

Pouring water

Make the three squares into equally deep water tanks attached to the sides of a right triangle.

Fill a² and b², then slowly tilt the board so the water runs into c². Guess first: will it fill up? Overflow?

Pythagorean triples

A right triangle with whole-number sides gives a Pythagorean triple, such as 3, 4, 5 or 5, 12, 13.

For any whole numbers m > n, a = m² − n², b = 2mn, c = m² + n² is a triple. Drag m and n, or click points on the plot.