Similarity: Length, Area and Volume Scale Factors - Worksheets, Questions and Revision

14 original exam-style questions - 4 pages of questions with a full mark scheme - free printable PDF.

Download PDF Jump to mark scheme (page 5)Similarity: Length, Area and Volume Scale Factors revision guide

This topic is chapter 4 of IGCSE Maths Practice Book 2.

Next in this strand

3.3 Similarity: Length, Area and Volume Scale Factors

EDEXCEL 4MA1 · Calculator allowed · about 55 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer in full sentences for any question that asks to 'Show that' or to 'Explain'. Spend about 60 minutes on the paper.

Two similar rectangles A and B have corresponding lengths 5 cm (in A) and 15 cm (in B). Calculate the linear scale factor from rectangle A to rectangle B.

(Total for Question 1 is 1 mark)

Two squares on a sheet of paper represent similar shapes. The side of the smaller square is 6 cm and the side of the larger square is 9 cm. Find the linear scale factor from the smaller square to the larger square.

(Total for Question 2 is 1 mark)

Two equilateral triangles are similar. The side of the small triangle is 4 cm and the side of the large triangle is 10 cm. Calculate the area scale factor from the small triangle to the large triangle.

(Total for Question 3 is 2 marks)

Two similar regular pentagons have corresponding areas 49 cm² (small) and A cm² (large). The linear scale factor from small to large is 3. Calculate A, the area of the larger pentagon.

(Total for Question 4 is 2 marks)

A triangular lamina is similar to another triangular lamina. The larger has area 162 cm² and the smaller has area 72 cm². Calculate the linear scale factor from the smaller lamina to the larger lamina. Give your answer in simplest fractional form.

(Total for Question 5 is 3 marks)

A model car is made at a linear scale of 1:18 of the real car. The real car has a fuel tank capacity of 54 litres. Estimate the capacity of the model car in millilitres. Give your answer to the nearest millilitre.

(Total for Question 6 is 2 marks)

A cone and a similar cone have heights 8 cm and 20 cm respectively. Find the volume scale factor from the smaller cone to the larger cone.

(Total for Question 7 is 2 marks)

Diagram: Two similar right-angled triangles P and Q. Triangle P has legs 3 cm and 4 cm; triangle Q has a short leg corresponding to the 3 cm leg equal to 9 cm. Show that the triangles are similar, and find the length of the other leg in triangle Q.

(Total for Question 8 is 3 marks)

Two similar cones have volumes in the ratio 1:27. The smaller cone has height 5 cm. Find the height of the larger cone.

(Total for Question 9 is 3 marks)

Two similar solid wooden beams are used to make benches. The small beam has cross-sectional area 36 cm² and length 1.2 m; the large beam is similar with cross-sectional area 81 cm². Calculate the length of the larger beam, stating the steps in your working.

(Total for Question 10 is 4 marks)

Two similar cylinders have radii 4 cm and 6 cm. Their heights are in the same ratio as the radii. The volume of the smaller cylinder is 201.06 cm³ (to two decimal places). Calculate the volume of the larger cylinder. Give your answer to 2 decimal places.

(Total for Question 11 is 3 marks)

Two similar rectangular swimming pools are modelled. The smaller pool is 10 m by 4 m and holds 80 000 litres. The larger pool is similar with short side 6 m. Calculate the capacity of the larger pool in litres. Give your answer to the nearest litre.

(Total for Question 12 is 4 marks)

A metal model of a statue is made similar to the original statue. The linear scale factor from model to original is 8. The model has mass 0.125 kg. Find the mass of the original statue, assuming mass scales in proportion to volume.

(Total for Question 13 is 3 marks)

A museum displays a pair of similar bronze models of a famous column. The smaller model has height 40 cm and volume 0.064 m³. The larger model has height 1.6 m. Use similarity to calculate the volume of the larger model in cubic metres. Show your method and give the answer to three significant figures.

(Total for Question 14 is 4 marks)

Mark scheme · 3.3 Similarity: Length, Area and Volume Scale Factors

Question 1

B1 scale factor = 15/5 = 3 cao
Answer: 3

Question 2

B1 linear scale factor = 9/6 = 3/2 or 1.5 cao
Answer: scale factor = 3/2 or 1.5

Question 3

M1 finds linear scale factor = 10/4 = 5/2
A1 area scale factor = (5/2)² = 25/4 cao
Answer: area scale factor = 25/4

Question 4

M1 uses area scale factor = 3² = 9
A1 A = 49 * 9 = 441 cm² cao
Answer: 441 cm²

Question 5

M1 uses area ratio = 162/72 = 9/4 after simplifying
M1 takes square root to find linear ratio = √9/4 = 3/2
A1 linear scale factor = 3/2 cao
Answer: 3/2

Question 6

M1 uses capacity scale factor = (1/18)³ = 1/5832 and converts 54 litres to ml (54 000 ml) then multiplies
A1 capacity approx = 54 000 / 5832 = 9.259... so 9 ml to nearest ml cao
Answer: 9 ml

Question 7

M1 finds linear scale factor = 20/8 = 5/2
A1 volume scale factor = (5/2)³ = 125/8 cao
Answer: volume scale factor = 125/8

Question 8

M1 identifies ratio of corresponding sides = 9/3 = 3 and states corresponding sides are in same ratio, showing similarity
M1 sets up proportion for the other leg: other leg = 4 * 3
A1 other leg = 12 cm cao
Answer: 12 cm

Question 9

M1 uses volume ratio = (linear ratio)³, so linear ratio = cube root of 27/1 = 3
M1 sets up height scale: larger height = 5 * 3
A1 height = 15 cm cao
Answer: 15 cm

Question 10

M1 finds area ratio = 81/36 = 9/4 and uses linear ratio = √9/4 = 3/2
M1 states that lengths scale by 3/2
M1 multiplies length: 1.2 * 3/2
A1 length = 1.8 m cao
Answer: 1.8 m

Question 11

M1 finds linear scale factor = 6/4 = 3/2
M1 uses volume scale factor = (3/2)³ = 27/8 and multiplies 201.06 by 27/8
A1 volume = 201.06 * 27/8 = 678.5775... so 678.58 cm³ to 2dp cao
Answer: 678.58 cm³

Question 12

M1 finds linear scale factor = 6/4 = 3/2 (short side ratio)
M1 uses volume/capacity scale factor = (3/2)³ = 27/8
M1 multiplies capacity: 80 000 * 27/8
A1 capacity = 80 000 * 27/8 = 270 000 litres cao (to nearest litre)
Answer: 270000 litres

Question 13

M1 uses volume/mass scale factor = 8³ = 512
M1 multiplies mass: 0.125 * 512
A1 mass = 64 kg cao
Answer: 64 kg

Question 14

M1 finds linear scale factor from small to large = 1.6 / 0.40 = 4
M1 uses volume scale factor = 4³ = 64
M1 multiplies volume: 0.064 * 64
A1 volume = 4.10 m³ to 3 s.f. cao
Answer: 4.10 m³ (to 3 s.f.)

Similarity: Length, Area and Volume Scale Factors - Worksheets, Questions and Revision

Next in this strand

Related topics

3.3 Similarity: Length, Area and Volume Scale Factors

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Mark your answers

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14