Why the Pythagorean theorem works: a rearrangement proof

0:00This is a right triangle: one of its angles is a right angle, 90 degrees.

0:06The two shorter sides are the legs, a and b. The longest side, opposite the right angle, is the hypotenuse, c.

0:16In our example, a is 3 centimetres and b is 4 centimetres.

0:22The Pythagorean theorem says that in a right triangle, a squared plus b squared equals c squared, where a squared means a times a.

0:32Why is it true? Put four copies of the triangle in the corners of a big square.

0:38Each side of the big square is a plus b: 3 plus 4, or 7 centimetres.

0:45The gap in the middle has four sides of length c and four right angles: a square with area c squared.

0:53With our numbers, the big square is 7 times 7, or 49 square centimetres.

1:00Each triangle has area 3 times 4, divided by 2, or 6. All four cover 4 times 6, or 24.

1:10So the gap, c squared, is 49 minus 24, or 25 square centimetres.

1:18Now slide three of the triangles to new places, without turning them.

1:23Now the gap is two squares, one with side a and one with side b.

1:29Their areas are a squared, 3 times 3, or 9, and b squared, 4 times 4, or 16: 25 in total, the same as before.

1:40The big square and the four triangles did not change, so the gap has the same area both ways.

1:47So a squared plus b squared equals c squared. Since 5 times 5 is 25, c is 5 centimetres.

1:57Recap: moving the same triangles turns a gap of c squared into a squared plus b squared, so the two are equal.