Why Do Two Squares Make a Third? (The Pythagorean Theorem)

Almost everyone can recite a² + b² = c². Almost nobody can say why adding two squares together gives you a third one. This is the why, and it is not an equation about lengths at all — it is an equation about area. Draw a square on each side of a right triangle. The claim is that the two small ones hold exactly as much space as the big one. So we go and check: lean the square on the short side over until it is a tall slanted column, slide it down into the big square, and do the same with the other one. Neither shape gains or loses a drop of area while it moves — you can watch the number sit at 9.00 the whole way — and when both have landed, the big square is exactly full. Then we test it with real numbers (3, 4 and 5), open the corner past ninety degrees and watch the two squares fall twelve units short, squeeze it under ninety and watch them spill over by the same twelve, and finish with the two places people actually use it: a builder squaring a foundation, and a ship working out how far it is from home. No memorising. No prior knowledge assumed. Every number on screen is computed from the shape being drawn — the areas are measured off the polygons, the counts are the cells you can see, and the side lengths at every angle come from the law of cosines. Nothing is asserted. CHAPTERS 0:00 The equation everyone can recite 0:26 It was never about lengths — it's about area 0:49 A square on every side 1:11 The claim, and the column 1:38 Shear one: same base, same height 2:00 Shear two 2:20 They fill it exactly — the proof 2:39 Real numbers: 3, 4 and 5 3:00 Nine plus sixteen is twenty five 3:22 What if the corner isn't square? 3:55 Using it backwards 4:12 Builders 4:31 A ship, thirty miles east 4:45 A truth about space THE FIGURE USED THROUGHOUT a = 3, b = 4, c = 5. a² = 9, b² = 16, c² = 25. The hypotenuse runs horizontally with the big square below it, so the altitude dropped from the right angle cuts that square into a left part and a right part. The left part is 1.8 × 5 = 9. The right part is 3.2 × 5 = 16. That is what the two small squares shear and slide into, and it is why nothing has to be taken on trust: 9 + 16 = 25. A shear is a slide — hold one edge still and push the opposite edge along the line it already sits on. The base doesn't move and the height doesn't change, so neither does the area. It's the same fact as a deck of cards pushed over at an angle: still the same deck. WHERE THIS PROOF COMES FROM This is the shearing presentation of Euclid, Elements, Book I, Proposition 47 — the one that splits the square on the hypotenuse into the two rectangles. And a note the video itself doesn't make room for: the result is much older than the name attached to it. Babylonian scribes were using these triples by about 1800 BC (the tablet Plimpton 322), and it appears independently in Indian and Chinese mathematics long before any Greek proof survives. There is no solid evidence Pythagoras proved it himself; the attribution comes from writers several centuries later. Animated procedurally in code — every frame, and the score, generated rather than drawn. pythagorean theorem explained · pythagoras theorem proof · why is a squared plus b squared equal c squared · visual proof of the pythagorean theorem · pythagorean theorem without memorising · shearing proof · euclid book 1 proposition 47 · area proof pythagorean theorem · 3 4 5 triangle · right triangle explained · pythagorean theorem for beginners · what does a2+b2=c2 mean · pythagorean theorem animation · converse of the pythagorean theorem · how builders square a corner