Why a² + b² = c²: 3 Picture Proofs (Easiest First)

Step by step

1. A 60-inch TV, only 48 inches wide?
   Why is a sixty-inch TV only forty-eight inches wide?
2. Measured corner to corner · 36 inches tall
   Because TVs are measured corner to corner. This one is thirty-six inches tall.
3. Square corner: short sides a and b, slanted side c
   Look at the triangle under the corner line. It has a square corner. Call the short sides a and b, and the slanted side c.
4. 36² + 48² = 1296 + 2304 = 3600 = 60²
   The famous rule says a squared plus b squared equals c squared. Thirty-six squared plus forty-eight squared is sixty squared. So it's sixty inches!
5. Known worldwide · hundreds of proofs
   This rule has names all over the world, and hundreds of proofs. Let's watch three of them, easiest first.
6. Proof 1: four triangles in a big square
   Proof one. Put four copies of our triangle in the corners of a big square. Each side of the big square is a plus b.
7. The middle space = c²
   The space in the middle has four pink sides. So it's a tilted square, with area c squared.
8. Same square, same 4 triangles, new spots
   Now take a second big square, the same size. Put the same four triangles in, but pair them up into two rectangles.
9. Take away the same triangles → c² = a² + b²
   Now the space left over is an a by a square and a b by b square. Same big square, same four triangles. Take the triangles away, and what's left must match. So a squared plus b squared equals c squared!
10. Proof 2: Zhao Shuang's picture (China)
    Proof two is a picture by Zhao Shuang from China, about eighteen hundred years ago. Four triangles sit around a small square, inside a tilted square with pink sides.
11. Swing two triangles down
    Swing this triangle down. And swing this one down.
12. Same 5 pieces. Why two squares? Measure with colors
    All five pieces are still here, even the little square. But why do they make two squares? Let's measure with the colors.
13. Little square side = b − a · top: b − (b − a) = a → a × a
    The left side is a yellow a. In Zhao's picture, a blue b was a yellow a plus the little square's side, so that side is b minus a. Along the top, take it away from b, and a is left. So the left part is a by a.
14. Bottom: b + a − a = b · right side b → b × b
    The bottom is a blue b plus a yellow a. The left part uses a, so b is left, and the right side is a blue b. So the right part is b by b. a squared plus b squared equals c squared!
15. Cut b² along the c square's directions → 4 pieces
    Proof three is a puzzle by Henry Perigal from England, about a hundred and fifty years ago. Cut the b square through its middle, lined up with the sides of the c square. That makes four pieces.
16. 4 pieces + the a square = the c square
    Now slide the pieces around the a square. They fill the c square perfectly!
17. Square corner → a² + b² = c²
    Three proofs, one rule. For any triangle with a square corner, a squared plus b squared equals c squared.