Pythagoras in three dimensions

Finding diagonals and slant lengths in rectangular prisms, cubes, pyramids and cones by using Pythagoras' theorem twice.

Use Pythagoras twice

To find the longest rod that fits in a box, first find the diagonal of the base, then use it as one side of a vertical right-angled triangle.

Space diagonal of an l × w × h box

$$d = \sqrt{l^2 + w^2 + h^2}$$

Example 1●●

Find the space diagonal of a $12 \times 4 \times 3$ box.

Base diagonal: $\sqrt{144 + 16} = \sqrt{160}$

$d = \sqrt{160 + 9} = \sqrt{169} = 13$

Example 2●●

A cone has radius $5$ cm and height $12$ cm. Find its slant height.

$s = \sqrt{5^2 + 12^2} = 13$ cm

Flashcards

Box diagonal

$\sqrt{l^2 + w^2 + h^2}$

Cone slant height

$\sqrt{r^2 + h^2}$

Cube of side 1 diagonal

$\sqrt{3}$

Practice

Space diagonal of a $12 \times 4 \times 3$ box?
Slant height of a cone with radius $5$ and height $12$?
Space diagonal of a cube with side $2$ (2 decimal places)?
Diagonal of the base of a $4 \times 3$ box?
A $2 \times 4 \times h$ box has space diagonal $\sqrt{101}$. Find $h$.
A cone has slant height $10$ and radius $6$. Find its height.
Space diagonal of a $2 \times 3 \times 6$ box?
A square pyramid has base side $10$ and slant height (apex to the middle of a base edge) $13$. Find its vertical height.