# Similar‌ ‌Shapes‌

Here‌ ‌we‌ ‌will‌ ‌learn‌ ‌about‌ ‌similar‌ ‌shapes‌ ‌in‌ ‌maths,‌ ‌including‌ ‌what‌ ‌they‌ ‌are‌ ‌and‌ ‌how‌ ‌to‌ ‌identify‌ ‌similar‌ ‌shapes.‌ ‌We‌ ‌will‌ ‌also‌ ‌solve‌ ‌problems‌ ‌involving‌ ‌similar‌ ‌shapes‌ ‌where‌ ‌the‌ ‌scale‌ ‌factor‌ ‌is‌ ‌known‌ ‌or‌ ‌can‌ ‌be‌ ‌found.‌

There‌ ‌are‌ ‌similar‌ ‌shapes‌ ‌worksheets‌ ‌based‌ ‌on‌ ‌Edexcel,‌ ‌AQA‌ ‌and‌ ‌OCR‌ ‌exam‌ ‌questions,‌ ‌along‌ ‌with‌ ‌further‌ ‌guidance‌ ‌on‌ ‌where‌ ‌to‌ ‌go‌ ‌next‌ ‌if‌ ‌you’re‌ ‌still‌ ‌stuck.‌

## What are similar shapes?

Similar shapes are enlargements of each other using a scale factor.

All the corresponding angles in the similar shapes are equal and the corresponding lengths are in the same ratio.

E.g.

These two rectangles are similar shapes.

The scale factor of enlargement from shape A to shape B is 2 .

The angles are all 90^o

The ratio of the bases is 2:4 which simplifies to 1:2

The ratio of the heights is also 1:2

E.g.

These two parallelograms are similar shapes.

The scale factor of enlargement from shape A to shape B is 3.

The corresponding angles are all equal, 45^o and 135^o .

The ratio of the bases are 3:9 which simplifies to 1:3

The ratio of the perpendicular heights is also 1:3

### Scale factor for length, area and volume

The scale factors for length, area and volume are not the same.

In Higher GCSE Maths similar shapes are extended to look at area scale factor and volume scale factors

To work out the length scale factor we divide the length of the enlarged shape by the length of the original shape.

To work the area scale factor we square the length scale factor.

To work the volume scale factor we cube the length scale factor.

E.g.

• Comparing length A and length B we can work out the scale factor to be 3 .

• Comparing area A and area B we can work out the scale factor to be 9 .
This is the same as 3^2 .

• Comparing volume A and volume B we can work out the scale factor to be 27 .
This is the same as 3^3 .

Step-by-step guide: Scale factor

## How to decide if shapes are similar

In order to decide if shapes are similar:

1. Decide which sides are pairs of corresponding sides.
2. Find the ratios of the sides.
3. Check if the ratios are the same.

### Related lessons on congruence and similarity

Similar shapes is part of our series of lessons to support revision on congruence and similarity. You may find it helpful to start with the main congruence and similarity lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

## Similar shapes examples

### Example 1: decide if shapes are similar

Are these shapes similar?

1. Decide which sides are pairs of corresponding sides.

The bases of the rectangles are a pair of corresponding sides.

The heights of the rectangles are a pair of corresponding sides.

2Find the ratios of the sides.

When writing the ratios the order is very important.

Here the ratio is length A : length B

The ratio of the bases is 1:2

The ratio of the heights is 2:4 which simplifies to 1:2

3Check if the ratios are the same.

The rectangles are similar shapes. The ratios for the corresponding lengths are the same 1:2.

The scale factor of enlargement for shape A to shape B is 2 .

### Example 2: decide if shapes are similar

Are these shapes similar?

Decide which sides are pairs of corresponding sides.

Find the ratios of the sides.

Check if the ratios are the same.

## How to find a missing length

In order to find a missing side in a pair of similar shapes:

1. Decide which sides are pairs of corresponding sides.
2. Find the scale factor.
3. Use the scale factor to find the missing length.

## Missing length examples

### Example 3: finding a missing length

Here are two similar shapes. Find the length QR.

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

### Example 4: finding a missing length

Here are two similar triangles. Find the length BC.

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

## How to find a missing length in a triangle

In order to find a missing side in a pair of triangles when you are not told that the triangles are similar:

1. Use angle facts to determine which angles are equal.
2. Redraw the triangles side by side.
3. Decide which sides are pairs of corresponding sides.
4. Find the scale factor.
5. Use the scale factor to find the missing length.

## Missing length in a triangle examples

### Example 5: finding a missing length in a triangle

Work out the value of x.

Use angle facts to determine which angles are equal.

Redraw the triangles side by side.

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

### Example 6: finding a missing length in a triangle

Work out the value of x.

Use angle facts to determine which angles are equal.

Redraw the triangles side by side.

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

## How to find an area or volume using similar shapes

In order to find an area or volume using similar shapes:

1. Find the scale factor.
2. Use the scale factor to find the missing value.

## Area or volume using similar shapes examples

### Example 7: finding an area or volume

These two figures are similar.

The area of shape A is 60 \; cm^2

Find the area of shape B:

Find the scale factor.

Use the scale factor to find the missing value.

### Example 8: finding an area or volume

These two shapes are similar.

The volume of shape A is 400 \; cm^3

Find the volume of shape B:

Find the scale factor.

Use the scale factor to find the missing value.

### Common misconceptions

• Take care with the order of ratios

Make sure that you are consistent with your ratios.

E.g.

In this example, always write the A value first, and then the B value.

1 : 3 and 2 : 6.

The ratios are equal, so these shapes are similar shapes.

• In most diagrams the diagrams are NOT drawn to scale

Often diagrams for questions involving similar shapes are NOT drawn to scale. So, use the measurements given, rather than measuring for yourself.

• Shapes can be similar but in different orientations

The second shape may be in a different orientation to the first shape. The shapes can still be similar.

E.g.

Here shape A and Shape B are similar.

Shape B is an enlargement of shape A by scale factor 2.

Here shape B has been rotated to make the similarity easier to see.

• Scaling up or down

If you are finding a missing length in the larger shape you can multiply by the scale factor.  The scale factor will be a number greater than 1 .

If you are finding a missing length in the smaller shape you can multiply by the scale factor, but the scale factor will be a number between 0 and 1.

### Practice similar shapes questions

1. Consider if these shapes are similar:

Yes – sides in ratio 1:4

No – sides in ratio 1:3 and 1:4

No – sides in ratio 1:3 and 1:2

Yes – sides in ratio 1:3

The shapes are similar as the ratio of the corresponding sides are the same.

The ratio of the bases is \;\; 3:9
the ratio of the heights is \; 1:3

2. Consider if these shapes are similar:

Yes – sides in ratio 2:1

Yes – sides in ratio 3:1

No – sides in ratio 2:1 and 1:2

No – sides in ratio 3:1 and 2:1

The shapes are similar as the ratio of the corresponding sides are the same.

The ratio of the short sides is \;\; 4:2
the ratio of the long sides is \quad 8:4

3. These shapes are similar. Find the value of x.

x=11

x=10

x=9

x=8

The ratio of the bases is \;\; 6:12

The scale factor of enlargement is 2

x=5 \times 2 =10

4. These shapes are similar. Find the value of x.

x=11

x=10

x=12

x=12.5

The ratio of the bases is \;\; 6:9

The scale factor of enlargement is 1.5

x=8 \times 1.5=12

5. Find the value of x.

x=11

x=13

x=16

x=9

Use the parallel lines to identify equal angles.

Then we can find pairs of corresponding sides.

The ratio of the corresponding sides is \;\; 4:12

The scale factor of enlargement is \; 3

x=3 \times 3= 9

6. Find the value of x.

x=8

x=9

x=10

x=11

Use the parallel lines to identify equal angles.

Then we can find pairs of corresponding sides.

The ratio of the corresponding sides is \;\; 9:6

The scale factor of enlargement is \; \frac{2}{3}

x=12 \times \frac{2}{3}= 8

### Similar shapes GCSE questions

1. Which shape is similar to shape X?

(1 mark)

Shape D

(1)

2. Triangles ABC and DEF are similar.

(a)  Write down the size of angle y

(b)  Work out the value of x

(3 marks)

(a)

y=56

(1)

(b)

x:11=4:8 the scale factor is \frac{1}{2}

(1)

x=11\times \frac{1}{2}=5.5

(1)

3.

ABC and AED are straight lines.
BE is parallel to CD.

AE = 7.5 \; cm
BE = 6.8 \; cm

Work out the length CD

(2 marks)

10.5\div 7.5=1.4

(1)

CD=6.8\times 1.4 = 9.52

(1)

## Learning checklist

You have now learned how to:

• Compare lengths using ratio notation and/or scale factors
• Solve problems with similar shapes using ratio notation and/or scale factors
• Solve problem with areas and volumes using ratio notation and/or scale factors (HIGHER)

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