## What you need to know

Two shapes are similar if they are the same shape but not necessarily the same size. Specifically, all of the angles in two similar shapes are identical, but their side-lengths can be different. Typically, 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. Usually, we will need to firstly find the scale factor between two similar shapes, and then use it to find a missing side-length.

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

Firstly, we must calculate the scale factor. To do so, recognise that the base of the smaller triangle is 5cm, whilst the base of the bigger one is 15cm. The scale factor is some value that, when multiplied by the side-lengths of the smaller shape, gives us the side-lengths of the bigger shape. So, if we divide the length of the base of the bigger triangle by the base of the smaller triangle, that will give us our scale factor:

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

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

$x=3\times3=9\text{ cm}$.

The idea of similar shapes and scale factors appears again when doing enlargements. In these cases, we are given a shape drawn on a pair of axes and told to enlarge it by a given scale factor about a centre of enlargement. Sort of like the opposite of the example we just saw. Let’s take a look.

Example: Enlarge shape ABCD below by scale factor 2 about the origin.

Once this is done, the ends of each line are the corners of your new shape. Join them up, and you get the result of enlarging ABCD by scale factor 2 about the origin. As these shapes are similar, they should be the same shape. Also, since the scale factor is 2, all of the sides of the shape should be twice as long as the smaller shape, which they are. Note: the scale factor tells you how big the shape will be, the centre of enlargement tells you where it will be.

So, we’ve got a shape, a scale factor, and a centre of enlargement – the origin. To perform this enlargement, we must draw a line from the centre of enlargement to each corner of the shape, ABCD. Then, since the scale factor is 2, we want to extend all of those lines to be 2 times as long (scale factor 3 would mean 3 times as long, and so on). For example, the line that goes from the origin to A is 2 across and 3 up. So, making that line twice as long means your resulting line should be 4 across and 6 up. Repeat this for all 4 corners.

You may come across fractional scale factors. The idea is the same, it’s just that the resulting shape will be smaller, not bigger. For example, if the scale factor is $\frac{1}{2}$, then you would draw the lines to each corner of the shape, as we did, and then you would you mark the halfway point along them/make them half as long. If you’re on the higher course, you will come across negative scale factors – in this case, you have to extend the lines to the other side of the centre of enlargement. See the third question at the bottom of the page for an example.

On the higher course, we extend the idea of similarity further to include not only side-lengths of shapes but also areas and volumes. The rules for this are as follows. If you have a scale factor, call it $SF$, that relates the side-lengths of two similar shapes, then

– The scale factor for the areas is $SF_A=(SF)^2$.

– The scale factor for the volumes is $SF_V=(SF)^3$.

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 by dividing the side-length of the bigger shape by that of the smaller shape: $SF=28\div7=4$.

Now, if the scale factor for the side-lengths is 4, then that means that the scale factor for the areas is: $SF_A=4^2=16$.

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

$\text{Area of A }=320\div16=20\text{ cm}^2$.

Example: A and B are similar shapes. Work out the volume of shape B.

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

$\text{Area of A }=320\div16=20\text{ cm}^2$.

## Example Questions

#### 1) Enlarge shape ABC by scale factor 3 about the point $(0, 1)$.

We need to draw lines from the point $(0, 1)$ to all corners of this shape. Then, since this is scale factor 3 enlargement, we need to extend these lines until they are 3 times longer. For example, the line from $(0, 1)$ to A goes 1 space to the right and 1 up. So, once we’ve extended it, the resulting line should go 3 spaces to right and 3 spaces up.

Then, once all these lines have been drawn, their ends will be the corners of the enlarged shape. Joining these corners up, we get the completed shape, as seen below.

#### 2) Shapes P and Q are similar.a) Calculate the size of missing side-length $x$.b) (HIGHER ONLY) Calculate the area of shape Q.

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

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

#### 3) (HIGHER ONLY) Enlarge shape ABC by scale factor -1 about the origin.

How to enlarge with a negative scale factor is a little less intuitive, but it’s not much more difficult. We still start by drawing lines from the centre of enlargement – here, the origin – to each corner of the shape. Now, rather than extending the lines outward from the corner, we extend the lines past the centre of the enlargement.

Because the scale factor is -1, the extension part of the lines (the part that goes outward from the origin, away from the shape) will be the same length as the original lines that were drawn from the corners to ABC. If the scale factor were -2, then the extension part of the lines would be twice the length of the original lines. This is subtly different to positive scale factors, so make sure you understand it.

For example, the line from the origin to C goes 2 to the right and 1 up. So, the extension to this line will, from the origin, go 2 to the left, and 1 down. Carrying this on with all the points, and then joining up the ends of the lines (since they form the corners of our shape), we get

If you have a keen eye, you’ll notice this is actually equivalent to rotating the shape around the centre of enlargement by $180\degree$. Neat.

You may be a GCSE Maths tutor in Harrogate or a trainee teacher in London and you need some new resources for a GCSE Maths class you are about to teach. Whether you are teaching a class or an individual student, the similar shapes revision worksheets on this page will come in handy. From straight forward questions to reverse scale factor questions, the similar shape revision materials presented cover many different question types suitable for many different abilities. For more GCSE Maths resources visit our main page.