GCSE Maths · Higher · Chapter 12: Similarity

Similar triangles, and the area and volume of similar shapes

Grades 4–9About 80 minutesCalculator allowed90 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Write the ratio $15 : 24$ in its simplest form.
[1 mark]
Answer

A2
Work out $2. 5^{3}$ .
[1 mark]
Answer
A3
Work out $3343$ .
[1 mark]
Answer
A4
Solve $\frac{x}{6} = \frac{9}{4}$ .
[2 marks]
Answer
A5
Two angles of a triangle are $4 9^{\circ}$ and $7 3^{\circ}$ . Work out the third angle.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Triangles $A B C$ and $P QR$ are similar. $A B$ corresponds to $P Q$ and $B C$ corresponds to $QR$ .
$A B = 6$ cm, $P Q = 15$ cm and $B C = 8$ cm. Work out the length of $QR$ .

Hint Find the scale factor from the pair of sides you know both of:

15 \div 6

. Multiply the other side by it.

B1
The two triangles are similar. Work out the length $x$ .
[2 marks]
Not drawn accurately
Answer

Example 2

In triangle $A D E$ , $B$ is on $A D$ and $C$ is on $A E$ . $B C$ is parallel to $D E$ .
$A B = 4$ cm, $A D = 10$ cm and $B C = 6$ cm. Work out the length of $D E$ .

Hint

B C

parallel to

D E

makes triangle

A B C

similar to triangle

A D E

. Compare

A B

with the WHOLE side

A D

, not with

B D

B2
$B C$ is parallel to $D E$ . Work out the length $x$ .
[3 marks]
Not drawn accurately
Answer

Example 3

Two shapes are similar. A side of the smaller shape is 3 cm and the matching side of the larger shape is 7.5 cm. The area of the smaller shape is 24 cm². Work out the area of the larger shape.

Hint The length scale factor is

2.5

. Areas are multiplied by the square of it.

B3
Two triangles are similar. A side of the smaller triangle is 6 cm and the matching side of the larger triangle is 9 cm. The area of the larger triangle is 81 cm².
Work out the area of the smaller triangle.
[3 marks]
Answer

Example 4

Two bottles are similar. Their volumes are 250 cm³ and 2000 cm³. The smaller bottle is 12 cm tall. Work out the height of the larger bottle.

Hint The volume scale factor is

2000 \div 250 = 8

. The length scale factor is the cube root of it.

B4
Two cones are similar. Their volumes are 135 cm³ and 625 cm³. The larger cone is 20 cm tall.
Work out the height of the smaller cone.
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Triangle $P$ has angles of $4 8^{\circ}$ and $7 7^{\circ}$ . Triangle $Q$ has angles of $7 7^{\circ}$ and $5 5^{\circ}$ .
Show that the two triangles are similar.
[2 marks]

C2
$A B$ is parallel to $D E$ . The lines $A E$ and $B D$ cross at $C$ .
Work out the length $x$ .
[2 marks]
Not drawn accurately
Answer
C3
$B C$ is parallel to $D E$ .
Work out the length of $B D$ .
[3 marks]
Not drawn accurately
Answer
C4
Two triangles are similar. Their areas are 32 cm² and 72 cm². The shortest side of the smaller triangle is 6 cm.
Work out the shortest side of the larger triangle.
[3 marks]
Answer
C5
Two solids are similar. Their heights are 5 cm and 8 cm. The volume of the smaller solid is 75 cm³.
Work out the volume of the larger solid.
[2 marks]
Answer
C6
Two cylinders are similar. Their volumes are 54 cm³ and 250 cm³. The surface area of the smaller cylinder is 90 cm².
Work out the surface area of the larger cylinder.
[3 marks]
Answer
C7
A model of a van is made to a scale of $1 : 40$ .
(a)
The model is 11 cm long. Work out the length of the real van in metres.
[1 mark]
Answer
(b)
The windscreen of the real van has an area of 1.28 m². Work out the area of the windscreen on the model in cm².
[2 marks]
Answer
(c)
The load space of the model has a volume of 1.5 cm³. Work out the volume of the load space of the real van in m³.
[2 marks]
Answer
Total for C7: 5 marks
C8
A post 1.6 m tall casts a shadow 2.4 m long. At the same time, a tree casts a shadow 15 m long on the same level ground.
Work out the height of the tree.
[2 marks]
Answer
C9
Rectangle $R$ is 6 cm by 9 cm. Rectangle $S$ is 10 cm by 14 cm.
Show that the two rectangles are not similar.
[2 marks]

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
The two triangles are similar. Matching angles are marked the same.
Not drawn accurately
(a)
Work out the length $x$ .
[2 marks]
Answer
(b)
Work out the length $y$ .
[2 marks]
Answer
Total for D1: 4 marks

D2
Jo says, "The sides of shape $B$ are twice as long as the sides of shape $A$ . Shape $A$ has an area of 18 cm², so shape $B$ has an area of $2 \times 18 = 36$ cm²."
Shapes $A$ and $B$ are similar. Explain why Jo is wrong, and work out the area of shape $B$ .
[3 marks]
Answer
D3
$A B$ is parallel to $D E$ . The lines $A E$ and $B D$ cross at $C$ .
$A B = 4$ cm, $D E = 6$ cm, $A C = x$ cm and $C E = (x + 3)$ cm.
Work out the value of $x$ . You must show your working.
[3 marks]
Not drawn accurately
Answer
D4
Two jugs are similar. The smaller jug holds 0.4 litres and the larger jug holds 1.35 litres. The larger jug is 21 cm tall.
Work out the height of the smaller jug.
[3 marks]
Answer
D5
Two solids are similar. Their surface areas are 48 cm² and 108 cm². The volume of the smaller solid is 64 cm³.
Work out the volume of the larger solid.
[4 marks]
Answer
D6
$B$ is on $A D$ and $C$ is on $A E$ . $B C$ is parallel to $D E$ .
$A B = 6$ cm, $B D = 4$ cm and $B C = 9$ cm. The area of triangle $A B C$ is 27 cm².
Not drawn accurately
(a)
Work out the length of $D E$ .
[2 marks]
Answer
(b)
Work out the area of the shape $B D E C$ .
[4 marks]
Answer
Total for D6: 6 marks
D7
Two statues are similar. One is 30 cm tall and the other is 75 cm tall. The smaller statue needs 0.2 litres of paint to cover it.
Paint is sold in tins of 0.5 litres. How many tins are needed to paint the larger statue?
[3 marks]
Answer
D8
A solid cone is 12 cm tall and has a volume of 540 cm³. A small cone 4 cm tall is cut off the top, with the cut parallel to the base. The part that is left is called a frustum.
Work out the volume of the frustum.
[3 marks]
Not drawn accurately
Answer
D9
A shop sells two similar posters. One is 40 cm wide and the other is 60 cm wide. The price of a poster is in proportion to its area. The smaller poster costs £3.20.
The shop sells the larger poster for £7.50. Is this the correct price? You must show your working.
[3 marks]
Answer
D10
Triangle $A B C$ has a right angle at $C$ . $D$ is the point on $A B$ where $C D$ is perpendicular to $A B$ .
$A C = 8$ cm and $A B = 10$ cm.
Not drawn accurately
(a)
Explain why triangle $A C D$ is similar to triangle $A B C$ .
[1 mark]
(b)
Work out the length of $A D$ .
[3 marks]
Answer
Total for D10: 4 marks

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
A rectangle is cut in half, parallel to its shorter side. Each half is similar to the whole rectangle.
Work out the ratio of the longer side to the shorter side of the rectangle.
[3 marks]
Hint 1 · What to try
Call the shorter side 1 and the longer side $x$ . Write the two sides of one half.
Hint 2 · The first line
One half is $1$ by $\frac{x}{2}$ , so its longer side is 1 and its shorter side is $\frac{x}{2}$ .
Hint 3 · The full method
Similar means $\frac{x}{1} = \frac{1}{x /2}$ , so $x^{2} = 2$ and $x = 2$ .
Answer

E2
Two solids are similar. Their volumes are 250 cm³ and 432 cm³. The surface area of the larger solid is 44 cm² more than the surface area of the smaller solid.
Work out the surface area of the smaller solid.
[4 marks]
Hint 1 · What to try
Find the length scale factor from the volumes first.
Hint 2 · The first line
$\frac{432}{250} = 1.728 = 1. 2^{3}$ , so the area scale factor is $1. 2^{2} = 1.44$ .
Hint 3 · The full method
If the smaller surface area is $S$ , then $1.44 S - S = 44$ , so $0.44 S = 44$ and $S = 100$ .
Answer
E3
In triangle $A B C$ , $B C = 12$ cm. A line $D E$ , parallel to $B C$ , cuts the triangle into two parts of equal area: a triangle $A D E$ and the part below it.
Work out the length of $D E$ . Give your answer to 3 significant figures.
[3 marks]
Hint 1 · What to try
Triangle $A D E$ is similar to triangle $A B C$ . What fraction of the area of $A B C$ is it?
Hint 2 · The first line
The area scale factor from $A D E$ to $A B C$ is 2.
Hint 3 · The full method
So the length scale factor is $2$ , and $D E = \frac{12}{2} = 8.49$ cm.
Answer
E4
Two similar triangles have perimeters of 24 cm and 36 cm. The area of the smaller triangle is 20 cm².
Work out the total area of the two triangles.
[3 marks]
Hint 1 · What to try
The perimeter is a length, so the perimeters give the length scale factor.
Hint 2 · The first line
Length scale factor $36 \div 24 = 1.5$ , so the area scale factor is $2.25$ .
Hint 3 · The full method
The larger area is $20 \times 2.25 = 45$ , so the total is $20 + 45 = 65$ cm².
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can show that two triangles are similar, and find a missing length using the scale factor.
I can find lengths in triangles cut by a parallel line, including where two lines cross.
I can use the area scale factor $k^{2}$ and the volume scale factor $k^{3}$ , and work back from them to $k$ .

Answers and mark scheme

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