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# Surface area and volume of cubes

## 17.3 Surface area and volume of cubes

A cube is a solid with six identical square faces at right angles to each other.

### Calculating surface area of a cube

A cube unfolded into a net is made up of six identical squares.

## Worked Example 17.1: Calculating the surface area of a cube

Calculate the surface area of the cube.

### Determine the area of one square face and multiply by six.

A cube has six identical square faces. We must calculate the area of one face and multiply it by six to get the total surface area of the cube.

\begin{align} \text{Surface area} &= 6 \times \text{area of square face} \\ &= 6 \times (9 \times 9) \\ &= 6 \times 81 \\ &= 486 \end{align}

Give the answer using the correct units. The diagram gives the measurement in metres, and the units of surface area are square units.

$\text{Surface area of cube} = 486 \text{ m}^2$

## Worked Example 17.2: Finding the length of cube from the surface area

The surface area of the cube is $$294 \text{ cm}^2$$. Find the length $$x$$ of the cube.

### Determine the length of the cube using formula for surface area.

\begin{align} \text{Surface area} &= 6 \times \text{area of square face} \\ 294 &= 6 \times (x \times x) \\ 294 &= 6 \times x^2 \\ \frac{294}{6} &= \frac{6x^2}{6} \\ 49 &= x^2 \\ \sqrt{49} &= x \\ 7 &= x \end{align}

$\text{Length of cube } x = 7 \text{ cm}$
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### Calculating volume of a cube

We use this formula to calculate the volume of a cube:

\begin{align} \text{Volume of cube} &= \text{length} \times \text{breadth} \times \text{height} \\ &= l^3 \\ \end{align}

We can also express the formula for the volume of a cube as the product of the area of the base and the height of the cube.

$\text{Volume of cube} = \text{(area of the base)} \times \text{height}$

## Worked Example 17.3: Calculating the volume of a cube

Find the volume of the cube with $$l = 3 \text{ cm}$$.

### Write down the formula for the volume of a cube.

$\text{Volume} = l^3$

### Calculate the volume of the cube.

\begin{align} \text{Volume} &= l^3 \\ &= 3^3 \\ &= 27 \end{align}

The volume of the cube is $$27 \text{ cm}^3$$.