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WJEC GCSE Maths

Topic questions Ratio, proportion and rates of change Similarity, lengths, areas and volumes

Similarity, lengths, areas and volumes

24 questions · 95 marks · about 2 hours

Similarity, lengths, areas and volumes questions

TopicSimilarity, lengths, areas and volumes

WJEC 3300 Similarity, lengths, areas and volumes 24 questions

Unit 1: Non-calculator June 2025 Higher tier, Question 18 4 marks

Two similar solids have surface areas of 3 cm2 and 75 cm2.

The volume of the smaller solid is 0·4 cm3.

Calculate the volume of the larger solid.

[4]
Reveal mark scheme Mark scheme for question 18

Unit 2: Calculator-allowed June 2025 Higher tier, Question 6 2 marks

In the diagram below, the triangles ABC and DBE are mathematically similar.

= 8 cm, = 7.5 cm, = 5 cm and maths symbol = maths symbol. CB DB BE ACB DEB

Figure for question 6

Calculate the length of AB.

[2]
Reveal mark scheme Mark scheme for question 6

Unit 2: Calculator-allowed June 2024 Intermediate tier, Question 19, also on the Higher paper 4 marks

(a)

The triangles shown below are similar.

Figure for question 19

Calculate the value of x.

[2]
Reveal mark scheme Mark scheme for question 19(a)
(b)

Four rectangles labelled A, B, C and D are shown below.

Figure for question 19

Which two rectangles are similar?

Give a reason for your choice.

The two rectangles which are similar are rectangles........ and........

[2]
Reveal mark scheme Mark scheme for question 19(b)

Unit 2: Calculator-allowed June 2024 Higher tier, Question 7, also on the Intermediate paper 4 marks

(a)

The triangles shown below are similar.

Figure for question 7

Calculate the value of x.

[2]
Reveal mark scheme Mark scheme for question 7(a)
(b)

Four rectangles labelled A, B, C and D are shown below.

Figure for question 7

Which two rectangles are similar?

Give a reason for your choice.

The two rectangles which are similar are rectangles........ and........

[2]
Reveal mark scheme Mark scheme for question 7(b)

Unit 2: Calculator-allowed June 2024 Higher tier, Question 18 4 marks

Two similar prisms have volumes of 569 cm3 and 3821 cm3, as shown below.

The uniform cross-sectional area of the smaller prism is 29 cm2.

Figure for question 18

Calculate the uniform cross-sectional area of the larger prism.

[4]
Reveal mark scheme Mark scheme for question 18

Unit 1: Non-calculator November 2023 Intermediate tier, Question 15, also on the Higher paper 4 marks

Figure for question 15
(a)

Explain why Rectangle A is not mathematically similar to Rectangle B.

[2]
Reveal mark scheme Mark scheme for question 15(a)
(b)

Alun changes only one of the measurements of Rectangle B to make the two rectangles mathematically similar.

Write down a possible length and width of Alun’s new rectangle.

You must show all your working.

[2]
Reveal mark scheme Mark scheme for question 15(b)

Unit 1: Non-calculator November 2023 Higher tier, Question 7, also on the Intermediate paper 4 marks

Figure for question 7
(a)

Explain why Rectangle A is not mathematically similar to Rectangle B.

[2]
Reveal mark scheme Mark scheme for question 7(a)
(b)

Alun changes only one of the measurements of Rectangle B to make the two rectangles mathematically similar.

Write down a possible length and width of Alun’s new rectangle.

You must show all your working.

[2]
Reveal mark scheme Mark scheme for question 7(b)

Unit 2: Calculator-allowed November 2023 Higher tier, Question 14 3 marks

A solid has a height of 225 cm.

A larger, mathematically similar solid has a height of 855 cm. The surface area of the smaller solid is 5300 cm2.

Calculate the surface area of the larger solid.

[3]
Reveal mark scheme Mark scheme for question 14

Unit 2: Calculator-allowed June 2023 Higher tier, Question 10 8 marks

ABCDE is a regular pentagon with sides of length 11 cm. CDE is a sector of a circle with centre D and radius 11 cm. The two shapes are joined together, as shown below.

Figure for question 10
(a)

In this part of the question, you will be assessed on the quality of your organisation, communication and accuracy in writing.

Calculate the length of the arc CE shown in the diagram.

You must show all your working.

[6]
Reveal mark scheme Mark scheme for question 10(a)
(b)

In a shape similar to the one shown on the previous page, the regular pentagon has sides of length 671 cm.

Complete the following statement.

Total area of new shape = ………………………… × total area of original shape

You must show all your working.

[2]
Reveal mark scheme Mark scheme for question 10(b)

Unit 2: Calculator-allowed November 2022 Higher tier, Question 13 4 marks

Two similar shapes have perimeters of 241 cm and 719 cm. The area of the smaller shape is 2063 cm2.

Calculate the area of the larger shape.

Give your answer in m2.

[4]
Reveal mark scheme Mark scheme for question 13

Unit 1: Non-calculator June 2022 Intermediate tier, Question 15, also on the Higher paper 4 marks

In the diagram: • AB and ED are parallel • triangles ABC and DEC are similar.

Figure for question 15
(a)

Calculate the length of CE.

[2]
Reveal mark scheme Mark scheme for question 15(a)
(b)

Calculate the length of AB.

[2]
Reveal mark scheme Mark scheme for question 15(b)

Unit 1: Non-calculator June 2022 Higher tier, Question 5, also on the Intermediate paper 4 marks

In the diagram: • AB and ED are parallel • triangles ABC and DEC are similar.

Figure for question 5
(a)

Calculate the length of CE.

[2]
Reveal mark scheme Mark scheme for question 5(a)
(b)

Calculate the length of AB.

[2]
Reveal mark scheme Mark scheme for question 5(b)

Unit 2: Calculator-allowed June 2022 Higher tier, Question 17 3 marks

Solid A and Solid B are similar.

Solid A has a volume of 8000 cm3 and a height of 30 cm. Solid B has a volume of 4913 cm3.

Calculate the height of Solid B.

[3]
Reveal mark scheme Mark scheme for question 17

Unit 1: Non-calculator November 2021 Higher tier, Question 11 2 marks

Two similar solids are shown below.

Figure for question 11

Solid A has a height of 5 cm. Solid B has a height of 7 cm.

Mari claims that the surface area of solid B is more than double the surface area of solid A. Is Mari correct?

You must justify your answer.

[2]
Reveal mark scheme Mark scheme for question 11

Unit 1: Non-calculator November 2020 Higher tier, Question 13 3 marks

Two similar cones have volumes of 20 cm3 and 1280 cm3.

The radius of the base of the smaller cone is 2·3 cm.

Calculate the radius of the base of the larger cone.

[3]
Reveal mark scheme Mark scheme for question 13

Unit 2: Calculator-allowed November 2019 Higher tier, Question 11 9 marks

(a)
Figure for question 11
(a)(i)

Show that 2x2 + 3x – 50 = 0.

[3]
Reveal mark scheme Mark scheme for question 11(a)(i)
(a)(ii)

Solve the equation 2x2 + 3x – 50 = 0 to calculate the lengths AB and DC.

Give your answers correct to 1 decimal place.

Length of AB =……........ …….. cm Length of DC =……........ …….. cm

[4]
Reveal mark scheme Mark scheme for question 11(a)(ii)
(b)

A rhombus has an area of 36∙8 cm2. The rhombus is enlarged by a scale factor of 7.

Calculate the area of the enlarged rhombus.

[2]
Reveal mark scheme Mark scheme for question 11(b)

Unit 2: Calculator-allowed November 2018 Intermediate tier, Question 2 3 marks

Circle either TRUE or FALSE for each statement given below.

STATEMENT
All circles are similar.TRUE
FALSE
All squares are similar.TRUE
FALSE
All triangles are similar.TRUE
FALSE
All rectangles are similar.TRUE
FALSE
All regular hexagons are similar.TRUE
FALSE
[3]
Reveal mark scheme Mark scheme for question 2

Unit 2: Calculator-allowed June 2018 Intermediate tier, Question 17, also on the Higher paper 6 marks

The diagrams below show two similar shapes, ABCD and PQRS.

Figure for question 17
(a)

Calculate the value of x.

[2]
Reveal mark scheme Mark scheme for question 17(a)
(b)

Calculate the value of y.

[2]
Reveal mark scheme Mark scheme for question 17(b)
(c)

Explain clearly why the following statement cannot be true.

Question 17, shown as an image
[2]
Reveal mark scheme Mark scheme for question 17(c)

Unit 2: Calculator-allowed June 2018 Higher tier, Question 7, also on the Intermediate paper 6 marks

The diagrams below show two similar shapes, ABCD and PQRS.

Figure for question 7
(a)

Calculate the value of x.

[2]
Reveal mark scheme Mark scheme for question 7(a)
(b)

Calculate the value of y.

[2]
Reveal mark scheme Mark scheme for question 7(b)
(c)

Explain clearly why the following statement cannot be true.

Question 7, shown as an image
[2]
Reveal mark scheme Mark scheme for question 7(c)

Unit 2: Calculator-allowed November 2017 Higher tier, Question 15 3 marks

Two similar pyramids have volumes of 3970 cm3 and 3100 cm3 respectively. The height of the larger pyramid is 25 cm.

Calculate the height of the smaller pyramid.

[3]
Reveal mark scheme Mark scheme for question 15

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