Formula

Space diagonal of a cube and a cuboid

Mensuration • Solids • Cube

Formula

d = l^{2} + b^{2} + h^{2}

A cuboid with edges

l

b

and

h

has space diagonal

l^{2} + b^{2} + h^{2}

, obtained by applying Pythagoras first in a face to get the face diagonal

l^{2} + b^{2}

and then in the vertical plane through it. For a cube of edge

a

this is

a 3

, while a face diagonal is only

a 2

How to use it

1Given the space diagonal, recover the edge as $a = \frac{d}{3}$ before computing any surface area or volume; usually only $a^{2}$ is needed, so the surd squares away at once.
2Keep the two diagonals apart: $a 2$ runs across one face, $a 3$ through the solid.
3It is the length of the longest straight rod that fits inside the box.

Worked example

A cuboid measures

3

cm by

4

cm by

12

cm. Find the length of its longest diagonal.

Step-by-step

1
$face diagonal = 3^{2} + 4^{2} = 5$ cm
Why: Pythagoras in the $3$ by $4$ face.
2
$d = 5^{2} + 1 2^{2} = 169 = 13$ cm
Why: Pythagoras again, with the face diagonal and the height as the legs.

Answer

13

Watch out

Using the face diagonal

a 2

where the space diagonal is meant is the standard error. The formula needs the three edges to be mutually perpendicular, so it applies to a cuboid and not to a general parallelepiped.

Builds on

Pythagoras' theorem

Also known as

space diagonal, diagonal of a cube, diagonal of a cuboid, a root 3, longest rod in a box, body diagonal

Space diagonal of a cube and a cuboid

Formula

Worked example

Appears in past papers

2019 (I)

2018 (II)