Rectangular Prism Space Diagonal

Two applications of Pythagoras give the space diagonal of a box. First across the floor, $\sqrt{l^2 + w^2}$, then up to the far top corner, which folds into the tidy $d = \sqrt{l^2 + w^2 + h^2}$. The pattern continues into any number of dimensions, which is why this is really the Euclidean distance formula wearing a cardboard costume.

Movers and shippers use it constantly. A carton 12 in × 12 in × 30 in will take a rod up to $\sqrt{144 + 144 + 900} = 34.4$ in long, and airlines quoting a "linear inches" limit are measuring something different again, so do not confuse the two. Note the one real trap: the diagonal is not the sum of the edges and it is not the sum of the face diagonals. It is always shorter than you would guess by adding.

There are integer cases, called Pythagorean quadruples: 1-2-2 gives 3, and 3-4-12 gives exactly 13. They are handy for checking your arithmetic, and rarer than the familiar 3-4-5 triples of the plane.