# 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 *$\underline{27:125}$.