# Cuboid perimeters

Can you find the volume of a cuboid, given its perimeters?

## Problem

The 'perimeter' of a cuboid can be mea

Image

One 'perimeter' of this cuboid is shown.

If the 'perimeters' of a cuboid are 12 cm, 16 cm and 24 cm, what is the volume of the cuboid?

*This problem is adapted from the World Mathematics Championships*

## Student Solutions

**Answer**: $35\text{cm}^3$

Label the widths $a, b, c$

Image

This perimeter is equal to $a+b+a+b=2a+2b$

Image

This perimeter is equal to $2a+2c$

Similarly, the third perimeter will be equal to $2b+2c$

$$2a+2b=12\Rightarrow a+b=6\\

2a+2c=16\Rightarrow a+c=8\\

2b+2c=24\Rightarrow b + c = 12$$

**Solving by elimination**

$\quad\qquad a+b=\ 6\\

\underline{+\ \quad\quad a+c=\ 8\ \ }\\

\quad 2a+b+c=14$

$\quad 2a+b+c=14\\

\underline{-\ \quad\quad b+c=12\ \ }\\

\quad \qquad\quad \ 2a=\ 2$

$\therefore a=1$

So $b=5$, $c=7$, volume $=1\times5\times7=35\text{cm}^3$

**Solving by substitution**

$a+b=6\Rightarrow b=6-a\\

a+c=8\Rightarrow c=8-a\\

\begin{align}b + c = 12\Rightarrow &(6-a)+(8-a)=12\\

\Rightarrow &14-2a=12\\

\Rightarrow &a=1\end{align}$

So $b=5$, $c=7$, volume $=1\times5\times7=35\text{cm}^3$

**Adding all of the equations together**

$\quad\qquad a+b=\ 6\\

\qquad\quad a+c=\ 8\\

\underline{+\ \quad\quad b+c=12\ \ }\\

\ \ \ 2a+2b+2c=26$

$\therefore a+b+c=13$

$c=13-6=7$

$b=13-8=5$

$a=13-12=1$

So the volume is $1\times5\times7=35\text{cm}^3$.