Area and volume scale factors

​​In a nutshell

3D shapes that are similar have proportional side lengths that are enlarged by a scale factor. This can be applied to finding the area scale factor and volume scale factor of similar 3D shapes.

Recap: similarity

Two shapes are similar if they are the same shape, but different sizes. This means that their sizes are proportional to one another; there is a scale factor relating the sides of one of the similar shapes to the other.

Similar 3D shapes

Linear scale factor

3D shapes can also be similar; it means that they have the same angles and overall shape, but the lengths of the sides are different. This means that there is also a "linear scale factor" that relates the side lengths of one shape to the other.

Example 1

Two cubes - cube A and cube B - are similar to each other. The side lengths of cube A are $$4cm$$. The side lengths of cube B are $$6cm$$​. What is the linear scale factor between the two cubes?

To find the linear scale factor, divide the two side lengths:

​$$6\div4=1.5$$​​

The linear scale factor is $$\underline{1.5}$$.

Area and volume scale factors

The area scale factor is the scale factor that relates the areas of two similar shapes.

The volume scale factor is the scale factor that relates the volumes of two similar shapes.

The area and volume scale factors are found by squaring or cubing the linear scale factor.

Example 2

Using the same cubes as in the above example:

i) Work out the surface area of cube A and find the surface area of cube B using the area scale factor.

ii) Work out the volume of cube B and find the volume of cube A using the volume scale factor.

Part i)

The surface area of cube A is given to be:

$$6\times(4^2)=6\times16=96cm^2$$​​

The linear scale factor was found to be $$1.5$$. The area scale factor is found by squaring this:

Area scale factor $$=1.5^2=2.25$$​​

Find the surface area of cube B by multiplying the surface area of cube A by the area scale factor:

​$$96\times2.25=216cm^2$$​​

The surface area of the cube B is $$\underline{216 \space cm^2}$$.

Part ii)

The volume of cube B is given to be:

​$$6^3=216cm^3$$​​

The volume scale factor is found by cubing the linear scale factor:

$$1.5^3=3.375$$​​

To find the volume of cube A, divide the volume of cube B by the volume scale factor. This is because cube A is smaller than cube B, so it must have a smaller volume.

$$216\div3.375=64cm^3$$​​

The volume of cube A is $$\underline{64 \space cm^3}$$.

Similarity expressed with ratios

The linear scale factor between two similar shapes can also be expressed by a ratio of two corresponding lengths. To find the area and volume ratios, square or cube both sides of the linear ratio.

Example 3

Two similar shapes have their surface areas in the ratio $$9:25$$. What is the ratio of their volumes?

First, find the linear ratio by taking the square root of both sides of the area ratio:

​$$\sqrt{9}:\sqrt{25}=3:5$$​​

The linear ratio is $$3:5$$.

Then, find the volume ratio by cubing both sides of the linear ratio:

​$$3^3:5^3=27:125$$​​

The ratio of their volume is $$\underline{27:125}$$.