Proving the Pythagorean Theorem by Cutting Squares — a² + b² = c²

The most famous rule in geometry says that in any right-angled triangle, the square built on the longest side equals the two squares on the shorter sides added together: a² + b² = c². The school of Pythagoras proved it around 530 BC. This blueprint proves it the maker's way — with a knotted cord, three cut squares, and a dissection you can hold in your hands. Seeing the pieces of the two small squares fit exactly into the big one is a proof that needs no algebra, and it doubles as the ancient builder's test for a true square corner.

初心者

手順

State the theorem

In a right-angled triangle, name the two short sides a and b and the longest side (the hypotenuse) c. The theorem says the square on c has exactly the same area as the squares on a and b together: a² + b² = c². You will prove this by area, not algebra.

Make a right angle with a cord

Knot a loop of cord into twelve equal spaces. Peg it out as a triangle with sides of 3, 4 and 5 spaces; the corner between the 3 and 4 sides is a perfect right angle. This 3-4-5 trick is how builders have squared corners for thousands of years.

必要な工具:

Cotton Kitchen String

Lay out a right triangle

Using that right angle, mark a right triangle with legs 3 and 4 units on a board. Its hypotenuse comes out to exactly 5 units — a whole-number right triangle to make the areas easy to count.

このステップの材料:

Red Alder Board1 個

必要な工具:

Chalk Line

Cut the three squares

Cut a square on each side of the triangle: 3×3, 4×4 and 5×5. Rule each into unit squares — 9, 16 and 25 of them. These three squares are the whole proof, made physical.

必要な工具:

Hand Saw

Knife

Count the areas

Count: the small squares hold 9 and 16 unit squares, together 25 — exactly the number in the big square on the hypotenuse. 9 + 16 = 25 is a² + b² = c² in plain counting.

Prove it by dissection

Now cut the two smaller squares into pieces and lay them inside the largest square. They tile it exactly — no gaps, no overlaps. Because the pieces fit for reasons of shape, not luck, this works for every right triangle, not only 3-4-5.

必要な工具:

Knife

Test other triangles

Repeat with other right triangles: the square on the hypotenuse always equals the sum of the other two. Change the corner so it is no longer a right angle, and the fit fails — which is exactly why the same rule can TEST whether a corner is truly square.

See why it matters

This single relationship underlies distance, surveying, navigation and building ever since. Any time you find a straight-line distance from two measurements at right angles, you are using Pythagoras — a 2,500-year-old proof you just held in your hands.