Similar Shapes and Enlargements Worksheets | Questions and Revision

# Similar Shapes and Enlargements Worksheets, Questions and Revision

Level 6 Level 7

## What are similar shapes

Two shapes are similar if all the angles in match, but not necessarily the same size.

Specifically, all of the angles in two similar shapes are identical, but their side-lengths can be different.

We relate two similar shapes of different sizes with a scale factor – a fixed number that when multiplied by the side-lengths of the smaller shape gives us the side-lengths of the bigger shape.

Make sure you are happy with the following topics before continuing.

## Finding the scale factor

The first step for any similar shape question will be to find the scale factor

To do this we need to find two corresponding dimensions for the example below we will use lengths.

(Surface area and volume can also be used, these are HIGHER ONLY)

Example: $A$ and $B$ are similar shapes.

Find the scale factor from $A$ to $B$.

First we need to find two corresponding lengths.

We can see that the base of $A=2$ and $B=5$

To calculate the scale factor we divide the larger be the smaller:

$5 \div 2 = 2.5$

Scale factor= $2.5$

## Using similar shapes to find a missing length

Once the scale factor has been found it can be used to find missing lengths.

Example: Shapes A and B are similar. Find the missing side-length marked $x$.

Firstly, we must calculate the scale factor.

To do this we use the base’s as these are corresponding, dividing the bigger by the smaller

$\textcolor{red}{\text{Scale factor }}=15\div5=\textcolor{red}{3}$

Now we have that the scale factor is $\textcolor{red}{3}$, all we need to do to find $x$ is multiply $\textcolor{red}{3}$ by the length of the corresponding side on the smaller shape. So, we get

$x=\textcolor{red}{3}\times3=9$

$x=9$cm

## Similar shapes and Area

(HIGHER ONLY) We can extend the idea of similarity further to include areasand volumes.

The rules for for area is as follows.

Scale factor for the areas = $SF_A=(SF)^2$

To remember this, think of it like units. If we measure a side-length in metres, then we measure the area in metres squared, and the volume in metres cubed.

Example: $A$ and $B$ are similar shapes.

Work out the area of shape $A$.

Firstly, we will determine the scale factor that relates the side-lengths, dividing the larger by the smaller

$SF=28\div7=\textcolor{red}{4}$.

Now, if the scale factor for the side-lengths is $\textcolor{red}{4}$, then that means that the scale factor for the areas is:

$SF_A=\textcolor{red}{4}^2=\textcolor{blue}{16}$

Therefore, to find the area of the smaller shape, we need to divide the area of the bigger shape by the area scale factor: 16. Doing so, we get

$\text{Area of A }=320\div\textcolor{blue}{16}=20$

$x = 20 \text{ cm}^2$

## Similar shapes and Volume

(HIGHER ONLY) We can extend the idea of similarity further to include areas and volumes.

The rules for for Volume is as follows:

Scale factor for the volumes = $SF_V=(SF)^3$.

Example: $A$ and $B$ are similar shapes.

The volume of shape $A = 600\text{ cm}^3$

Work out the volume of shape $B$.

Firstly, we will determine the scale factor that relates the side-lengths, dividing the larger by the smaller

$SF=22\div11=\textcolor{red}{2}$.

Now, if the scale factor for the side-lengths is $\textcolor{red}{4}$, then that means that the scale factor for the areas is:

$SF_A=\textcolor{red}{2}^3=\textcolor{blue}{18}$

We know the volume of shape $A$ and the Volume scale factor $=18$.

So we need to multiply the volume of $A$ by the $SF^3$

Volume of $B = 600 \times \textcolor{blue}{18} = 10800\text{ cm}^3$

### Example Questions

a) To work out the scale factor, $SF$, we need to divide the given side-length on the bigger shape by the corresponding side of the smaller shape. Doing so, we get

$SF=5\div2=2.5$

Then, to find $x$, we must multiply this value by the corresponding side-length of the smaller shape. So, we get

$x=2.5\times3=7.5\text{ cm}$

b) To find the scale factor for the areas, $SF_A$, we must square the known scale factor:

$SF_A=2.5^2=6.25$

Now, to get the area of the bigger shape, we must multiply the area of the smaller one by this scale factor. Doing so, we get

$\text{Area of Q}=6\times6.25=37.5\text{ cm}^2$

a) To work out the scale factor, $SF$, we need to divide the given side-length on the bigger shape by the corresponding side of the smaller shape. Doing so, we get

$SF=42\div14=3$

b) Now we have the scale factor, we can apply it to the corresponding length to AC which is DF. Hence, we find that,

$\text{AC} =51\div3=17 \text{cm}$

a) To work out the scale factor, $SF$, we need to divide the given side-length on the bigger shape by the corresponding side of the smaller shape. Doing so, we get,

$SF=6\div6=2$

b) Now we have the scale factor, we can apply it to the corresponding length to BE which is BC. Hence, we find that,

$\text{AC} =4.4\times2=8.8 \text{cm}$

The surface area of a sphere is given by the formula,

$\text{surface area} =4\pi \text{r}^2$

The scale factor between the radius of the two similar spheres is a factor of 3.

Considering the ratio of surface areas, we find that,

$\text{SA of larger sphere : SA of smaller sphere} = 4\pi (3x)^2 : 4\pi x^2$

This simplifies to,

$\text{SA of larger sphere : SA of smaller sphere} = 9 : 1$

To work out the scale factor, $SF$, we need to divide the given side-length on the bigger shape by the corresponding side of the smaller shape. However, in this case we are not given two corresponding sides. Instead we can set an unknown length BE, as $x$ and form the equation,

$\dfrac{x}{5}=\dfrac{20}{x}$

Rearranging to find,

$x^2=100$

$x=10$

Now we have to corresponding sides that we can use to find the scale factor,

$SF=20\div10=2$

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