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how does pythagorean theorem work in 3d
How long a stick fits in a box 4 wide, 3 deep and 12 tall? The longest runs corner to opposite corner, straight through the middle: the 3D diagonal.
First flatten it. On the floor, the diagonal of the 3 by 4 rectangle is the longest side of a right triangle: 9 plus 16 is 25, so the floor diagonal is 5.
Now stand a second right triangle on that: the floor diagonal, 5, and the height, 12. Twenty-five plus 144 is 169, so the stick is 13 long.
Do it in letters. Call the floor sides a and b and the floor diagonal f. Nothing new yet: f squared equals a squared plus b squared, the plain old theorem.
Second use: the stick d and the height c with f. d squared equals f squared plus c squared. Replace f squared by a squared plus b squared, and the 3D theorem appears.
So the 3D diagonal is the square root of a squared plus b squared plus c squared. Our box: 13. A cube with side one: root three, about 1.73.
Three dimensions need no new theorem: walk across the floor, then up the wall, and Pythagoras does the rest.