Pythagoras by Scissors

What’s going on

Draw a right triangle, a triangle with one square corner. Call its two short sides a and b, and its long side, the one opposite the square corner, c. Now stand a square on each side.

The famous claim is that the two small squares together have exactly the same area as the big one:

a² + b² = c²

For the triangle in the scene, with sides 3 and 4: 9 + 16 = 25, so the long side is 5. But a formula with numbers is a check, not a reason. The scene shows the reason with scissors: you can cut the two small squares into pieces that fill the big square with nothing left over.

Cut the squares

Press Slide the pieces.

1. The square on the longer leg is cut through its centre by two lines at right angles: one runs the same way as the long side c, the other straight across it. That makes four pieces of the same shape.
2. The four pieces slide into the corners of the square on c. None of them turns. Each one only moves.
3. A square hole is left in the middle, exactly the size of the square on the shorter leg. That square slides in too.

Pieces that only slide keep their area, so a² + b² = c². Change a and b and slide again: it works for every right triangle, thin or fat. Try a = b, where the cuts go along the diagonals and the pieces become triangles.

Why do the pieces fit the corners so well? Each cut line crosses the square through its centre, at the same tilt as the long side, and a line like that crosses the square in a segment exactly c long. So every piece has two edges of length c ÷ 2 meeting at a right angle, just like a corner of the big square.

Move four triangles

Choose Move four triangles. Here are four copies of the triangle in a square whose side is a + b.

• Laid out like a pinwheel, the triangles leave a tilted square uncovered in the middle. Its side is c, so its area is c².
• Slide three of the triangles (the fourth stays where it is). Now the uncovered part is two squares, a² and b².

The big square did not change, and neither did the four triangles. So what is left over must be the same in both pictures: c² = a² + b².

Make it from paper

Draw a right triangle on squared paper, for example with legs 3 and 4 squares. Draw the three squares on its sides. Cut out the two small squares, cut the larger one through its centre as in the scene, and fit all five pieces into the square on the long side. Then try a triangle with legs 2 and 5: the square on the long side does not fit the grid lines any more, but the pieces still fill it.

Try it yourself

1. The ladder

A ladder 5 m long leans against a wall. Its foot is 3 m from the wall. How high up the wall does it reach?

Answer

4 m. The ladder is the long side of a right triangle, the ground and the wall are the short sides. 3² + h² = 5², so h² = 25 − 9 = 16 and h = 4.

2. Half a square

Cut a square with side 1 along its diagonal. How long is the diagonal?

Answer

√2 ≈ 1.414. The diagonal is the long side of a right triangle with both short sides 1, so its square is 1 + 1 = 2. Set a = b in the scene: the square on the diagonal is made of exactly two small squares.

3. A tilted square on graph paper

On squared paper, draw a square whose corners are grid points: start at one point, go 3 right and 1 up for the first side, then turn a right angle each time. What is its area, in grid squares?

Answer

10. Each side is the long side of a right triangle with short sides 3 and 1, so the area of the square is 3² + 1² = 9 + 1 = 10. You can check by counting: draw a 4 × 4 square around it, area 16, and take away the four corner triangles, each 3 × 1 ÷ 2 = 1.5: 16 − 6 = 10.

Go deeper

• The rule was known long before Pythagoras. A Babylonian clay tablet from about 1800 BC, called Plimpton 322, lists sets of whole numbers like 3, 4, 5 that fit it.
• The five-piece cut in the scene was found by Henry Perigal, a London amateur mathematician, in the 19th century. He liked it so much that the diagram is carved on his gravestone.
• There are hundreds of different proofs. One was found by James Garfield in 1876, a few years before he became President of the United States.