Nib asked
How does Pythagorean theorem work in 3D?
In three dimensions you use the ordinary Pythagorean theorem twice, once across the floor and once up the height, which gives d squared equals a squared plus b squared plus c squared.
6 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1the box
Picture a box 4 wide, 3 deep and 12 tall. The longest straight stick that fits runs from one bottom corner to the opposite top corner, through the inside of the box. That length is the space diagonal.
The longest stick in a box runs corner to opposite corner.
a common mix-upThere is no separate 3D Pythagorean theorem to memorize: it is the ordinary theorem used twice.
2across the floor
The floor of the box is a 3 by 4 rectangle. Its diagonal is the hypotenuse of a right triangle with legs 3 and 4, so it is the square root of 9 + 16 = 25, which is 5.
The floor diagonal of a 3 by 4 rectangle is 5.
3up the height
The floor diagonal (5) and the vertical edge (12) meet at a right angle, because the edge stands straight up from the floor. They form a second right triangle whose hypotenuse is the stick: the square root of 25 + 144 = 169, which is 13.
Floor diagonal 5 and height 12 give a stick of 13.
A box is 2 by 3 by 6. How long is its longest stick?
Show the answer
7, since 4 + 9 + 36 = 49 and the square root of 49 is 7.
4floor triangle
Call the floor sides a and b and its diagonal f. By the ordinary theorem, f squared equals a squared plus b squared.
f² = a² + b² on the floor.
5height triangle
The vertical edge c is perpendicular to the floor, so f and c form a right triangle with hypotenuse d: d squared = f squared + c squared. Substituting f squared = a squared + b squared gives d squared = a squared + b squared + c squared.
d² = a² + b² + c²: the theorem, used twice.
6the whole idea
The space diagonal of a box with sides a, b, c is the square root of a² + b² + c². For 3, 4, 12 it is 13. For a cube of side 1 it is the square root of 3, about 1.73.
d = √(a² + b² + c²).
Three dimensions need no new theorem: walk across the floor, then up the wall, and Pythagoras does the rest.