Diagonal of a cuboid calculator

A cuboid with edges a, b and c has three different faces, and so three different face diagonals, each of which you work out like the diagonal of a rectangle. The space diagonal runs through the whole cuboid, from one corner to the opposite one.

Cuboid with edges a, b and c and the red space diagonal d_s. It is the hypotenuse of a right-angled triangle whose two shorter sides are the blue diagonal of the bottom face and the edge c

The diagonals of the three faces of the cuboid: face a × b in blue, a × c in orange and b × c in green

How it works

We get the space diagonal by applying Pythagoras’ theorem twice:

$d_{s} = \sqrt{a^{2} + b^{2} + c^{2}}$

Will a 13 cm pencil fit inside a box measuring 3 × 4 × 12 cm? Not along any edge or face, since the longest face diagonal is about 12.6 cm. Diagonally across the whole box, though, it fits exactly: $\sqrt{9 + 16 + 144} = \sqrt{169} = 13$ .

If all the edges are the same length, the solid is a cube, and there is a separate calculator for that.

Chapters: Diagonal of a square calculator, Diagonal of a rectangle calculator, Diagonal of a cube calculator, Diagonal of a cuboid calculator, Cylindrical tank volume calculator