Cut the square, keep the area

Take a right triangle with legs a and b and hypotenuse c. The Pythagorean theorem says a² + b² = c². You can prove it with scissors: cut four copies of the triangle, lay them in a big square two different ways, and look at what is left over.

Why it is worth a look

A dissection proof turns the formula into something you can see. It needs no algebra beyond the area of a square, and every step is an action you can check with paper.

It also shows what an area proof rests on. Rachel Morley's 2018 undergraduate thesis at Boise State collects four area proofs of the theorem, among them a stacking proof credited to Bhaskara and a similar-triangles proof credited to Euclid, and frames the cut-and-rearrange ones as classroom activities. It states two rules about area: congruent triangles have equal area, and the areas of non-overlapping pieces add up to the whole. Those rules alone are not enough: the thesis also uses the fact that a triangle’s angles add to 180°, which is exactly what a curved surface spoils.

The two arrangements

This dissection is the thesis’s third proof, credited there to Pythagoras; the drawing, the slider and the numbers are this page’s own. Morley turns two triangles to make two rectangles; here they only slide. Start with a square whose side is a + b. Every arrangement below lives inside that same square.

Arrangement one: put one triangle in each corner, with its right angle in the corner. Each side of the big square is then split into a piece of length a and a piece of length b. The four long sides enclose a tilted region in the middle. Its sides each have length c. Its corners are right angles too. A triangle’s angles add to 180°, so its two sharp angles add to 90°. At each corner of the middle region, one sharp angle from each of two triangles sits on a straight line, so the corner takes the remaining 90°. So the middle is a square of area c².

Arrangement two: slide the same four triangles, without turning or flipping any, into two pairs. Each pair makes a rectangle a by b. Push one rectangle into the bottom-right corner and the other into the top-left. What is left uncovered is a square of side a in one corner and a square of side b in the other: area a² + b².

Why this counts as a proof

In both arrangements the big square is the same, so its area is too. The four triangles are congruent copies, so by the first rule they cover the same area both times. By the second rule, the big square's area equals the triangles' area plus whatever is left over. Take the same triangle area away from the same total and the leftovers must match. The leftover in arrangement one is c². The leftover in arrangement two is a² + b². So a² + b² = c².

Nothing depends on the particular triangle, which the slider lets you test. Try a very thin triangle: the tilted square grows toward the whole big square, and so does the bigger corner square.

In short

Four copies of a right triangle fit inside a square of side a + b in two ways. One way leaves a tilted square of side c; the other leaves squares of sides a and b. The same square minus the same triangles leaves the same area, so c² equals a² + b². Congruent pieces have equal area, non-overlapping pieces add, and a triangle’s angles sum to 180°.