Perigal's Dissection Proof of the Pythagorean Theorem

Perigal's dissection proves the Pythagorean theorem by cutting and sliding. Two cuts through the center of the larger leg square, one parallel and one perpendicular to the hypotenuse, divide it into four congruent pieces. Those four pieces and the smaller leg square fit exactly into the square on the hypotenuse, so a² + b² = c².

Henry Perigal, a London bookkeeper and amateur mathematician, published it in the 1870s and liked it so much that he printed it on his business cards. It makes a great classroom activity: five paper pieces, no algebra, and a result you can hold in your hand. This page shows how to make the cuts, why the pieces always fit, and the hidden tiling pattern that explains it.

How to make the cuts

Call the legs a and b, with b the longer one, and the hypotenuse c. Then:

1. Draw the squares on all three sides of the right triangle, outward from the triangle.
2. Find the center O of the square on the longer leg, where its diagonals cross.
3. Through O, draw a line parallel to the hypotenuse. It crosses two opposite sides of the square.
4. Through O, draw a line perpendicular to the first one. It crosses the other two sides.

The four pieces are congruent. A quarter turn about O carries the square onto itself, and because the cuts are perpendicular lines through O, it also carries each cut onto the other. So the quarter turn moves piece 1 onto piece 2, piece 2 onto piece 3, and so on.

Watch the pieces slide

Each piece moves to one corner of the square on the hypotenuse, with the right angle it had at O fitting into that corner. The small square then fills the hole left in the middle. Notice that nothing rotates. The cut edges are parallel or perpendicular to the hypotenuse, and so are the sides of the hypotenuse square, so every piece is already facing the right way and only needs to slide (a translation). Perigal's dissection is a famous example of a proof by translation alone.

Measuring a piece

To prove the pieces fit, you need their measurements. Two facts do all the work.

Fact 1: each cut is exactly as long as the hypotenuse

Start at O and walk parallel to the longer leg until you reach the side of the square: that is half the side, b/2. Then walk along the side to the point where the cut meets it. These two walks and the half-cut form the purple right triangle in the figure. Its angle at O is the angle between the cut and leg b, which equals the angle between the hypotenuse and leg b, because the cut is parallel to the hypotenuse. So the purple triangle is similar to the original triangle, with the side b/2 in place of b. Every side is halved: its other leg is a/2 and its hypotenuse, the half-cut, is c/2.

The same holds for the perpendicular cut, by the quarter-turn symmetry. So both cuts have length c and are halved by O. This uses only similar triangles, not the Pythagorean theorem, so nothing is assumed that we are trying to prove.

Fact 2: the outer edges are (b + a)/2 and (b − a)/2

The cut meets the side of the square a/2 from the side's midpoint. Measured from the nearest corner, that point is b/2 − a/2 away on one side and b/2 + a/2 on the other. So each piece has:

• a right angle at O between two edges of length c/2;
• a right angle at the old corner of the square between edges of length b + a over 2 and b − a over 2;
• two other angles that add up to 180°, since the angles of any four-sided figure add up to 360°.

Why it works: the proof

1. The pieces are congruent. The quarter turn about O maps the square and the pair of cuts onto themselves, so it carries each piece onto the next.
2. The corners fit. Slide each piece so its right angle at O sits in a corner of the hypotenuse square. Its two c/2 edges then run along the two sides from that corner and stop exactly at their midpoints. Two pieces meet at each midpoint and their angles there are the two angles that add up to 180°, so their inner edges carry on along one straight line, with no gap and no overlap.
3. The hole is an a-by-a square. Every inner edge came from a side of the larger square, so it is parallel to one of the legs. Along each side of the hole, the long edge of one piece runs past the short edge of its neighbor, which leaves a side of length b + a over 2 − b − a over 2 = a
4. The small square fills the hole. The arrangement looks the same after a quarter turn about the center of the hypotenuse square, so all four sides of the hole are equal and its corners are right angles. It is a square of side a, exactly the size of the square on the shorter leg.
5. Count the area. The five pieces fill the hypotenuse square without overlapping, so c² = (four pieces) + a² = b² + a²

The tiling behind the dissection

Perigal's cuts are not a lucky accident. Cover the floor with copies of the two leg squares in the staircase pattern known as the Pythagorean tiling (it is a common pattern for floor tiles; see the theorem in art and architecture). The pattern repeats when you shift it a distance c along the direction of the hypotenuse, or a distance c at right angles to it.

Now lay a grid of hypotenuse-sized squares over the tiling, with its corners at the centers of the large squares. The grid lines through each large square are exactly Perigal's two cuts. Each grid square contains one small square and four pieces of large squares. Because the tiling repeats with the same shifts as the grid, those four pieces are copies of the four different pieces of a single large square. Every cell of the grid is Perigal's dissection, repeated across the whole plane.

This also explains why the pieces only slide: moving from one large square to another in the tiling never involves a turn.

Henry Perigal

Henry Perigal (1801–1898) worked first as a clerk and then, for more than forty years, as a bookkeeper for a London stockbroker. Mathematics and astronomy were his passions. He was elected a Fellow of the Royal Astronomical Society in 1850 and served for decades as treasurer of the Royal Meteorological Society.

Dissections, cutting one figure into pieces that rearrange into another, were his special interest. He published his dissection of the Pythagorean squares in the Messenger of Mathematics in the early 1870s, in a short paper on geometric dissections, and later in his booklet Geometric Dissections and Transpositions (1891). He was proud of it: it appeared on his business cards, and it is engraved on his tombstone in the churchyard at Wennington, Essex, now part of London.

Other famous dissection proofs work differently. Bhāskara's proof and the Chinese hypotenuse diagram cut the hypotenuse square into four triangles and a small square, and the rearrangement proof moves four triangles inside a larger square. Perigal's is unusual in cutting only one leg square and moving its pieces without turning them. Leonardo da Vinci's proof takes yet another route, adding triangles instead of cutting, and the proofs overview compares them all.

Try it with paper

1. Draw a right triangle with clearly different legs, such as 6 cm and 10 cm. Graph paper helps. The hypotenuse is √136 ≈ 11.66 cm, rounded to the nearest hundredth.
2. Draw the three squares on the sides, outward from the triangle.
3. Find the center of the 10 cm square by drawing its diagonals lightly.
4. Draw the first cut through the center, parallel to the hypotenuse. It meets two opposite sides 3 cm (half the short leg) from their midpoints, one above and one below. Check by eye that it really is parallel to the hypotenuse.
5. Draw the second cut through the center at right angles to the first. It meets the other two sides 3 cm from their midpoints.
6. Cut out the four pieces and the 6 cm square, then arrange them on the hypotenuse square without turning any piece.