Pythagoras: why a² + b² = c²

0:01 A ladder leans against a wall. How long does the ladder need to be?

0:06 The wall, the ground and the ladder make a right-angled triangle.

0:11 Let's start small, with sides of three and four.

0:16 Build a square on each short side. Three times three is nine little squares. Four times four is sixteen.

0:27 Now the longest side. Its square holds twenty-five.

0:33 Nine plus sixteen makes twenty-five. The two small squares fill the big one exactly.

0:42 Is that just luck? Let's check with any right-angled triangle.

0:49 Put four copies inside a big square. The space left in the middle is the square on the longest side.

0:55 Now slide the triangles around. The space left is the two smaller squares. Same space, so they must be equal.

1:06 A squared, plus b squared, equals c squared.

1:11 Back to the ladder. Six metres up, eight metres along the ground. Thirty-six plus sixty-four is one hundred, so the ladder needs ten metres.

1:20 Square the two short sides, add them, and take the square root. That's Pythagoras.