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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.
Anfänger
2
Anweisungen
1
1
State the theorem
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.
2
2
Make a right angle with a cord
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.
Benötigte Werkzeuge:
Cotton Kitchen String3
3
Lay out a right triangle
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.
Materialien für diesen Schritt:
Red Alder Board1 StückBenötigte Werkzeuge:
Chalk Line4
4
Cut the three squares
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.
Benötigte Werkzeuge:
Hand Saw
Knife5
5
Count the areas
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.
6
6
Prove it by dissection
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.
Benötigte Werkzeuge:
Knife7
7
Test other triangles
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.
8
8
See why it matters
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.
Materialien
1- 1 StückPlatzhalter
You can swap these in
Can't get one of the materials? Swap it for an equivalent — these work just as well.
- Instead of Knife, try:
Gilder's Knife
Blunt Collection Knife
Sharp Knife (faca)
Small Trimming Knife
Sharp Pruning Knife
Sharp Knife - Instead of Hand Saw, try:
Portable Band Saw
Band Saw (9-inch, Benchtop)
Miter Box with Saw
Small Hand Saw
Jeweler's Saw - Instead of Chalk Line, try:
Fishing Line (Monofilament)
Recommended for this build
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Dowel RodUsed together and in similar builds
Hardwood BlockFrequently used with this build's materials
Binding RopeUsed together and in similar builds
AwlUsed together and in similar builds
Beech LumberUsed together and in similar builds
Iron NailsFrequently used with this build's materials
Hand AugerUsed together and in similar buildsSharp KnifeFrequently used with this build's materialsVerwandte Blueprints
Diese Blueprints teilen Wissen — Techniken, Materialien oder Prinzipien
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CC0 Gemeinfrei
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