4.11 Similarity: The Complete Exam Archive

Edexcel 4MA1 • 15 Questions • Foundation → Grade 9 Progression

Phase 1

Q1-5 • Grades 5-6

Phase 2

Q6-10 • Grades 7-8

Phase 3

Q11-15 • Grades 8-9

Foundation Anchor

Foundation skill: Direct division for LSF

Question 1[1 mark]Grade 5-6

Two triangles are similar. A side of length 2 cm in the smaller triangle corresponds to a side of length 6 cm in the larger triangle. Find the Linear Scale Factor (LSF).

Question 2[1 mark]Grade 5-6

If the Linear Scale Factor (LSF) between two similar shapes is 2, what is the Area Scale Factor (ASF)?

Question 3[2 marks]Grade 5-6

A model car is made to a scale of 1:10. The real car has a length of 4 m. What is the length of the model car in cm?

Question 4[2 marks]Grade 5-6

Two similar rectangles have areas where the smaller has area 10 cm². The Linear Scale Factor is 3. Find the area of the larger rectangle.

Question 5[2 marks]Grade 6

The volume of a small cube is 8 cm³. The volume of a larger similar cube is 64 cm³. Find the Linear Scale Factor.

Higher Standard

Chain conversion: ASF → LSF → VSF

Question 6[3 marks]Grade 7-8

Two similar solids have surface areas of 10 cm² and 40 cm². The volume of the smaller solid is 5 cm³. Find the volume of the larger solid.

Question 7[3 marks]Grade 7

Two similar bottles have heights 10 cm and 15 cm. The volume of the smaller bottle is 200 ml. Find the volume of the larger bottle.

Question 8[3 marks]Grade 7-8

A map has a scale of 1:50,000. An area measures 4 cm² on the map. What is the actual area in km²?

Question 9[2 marks]Grade 8

Two similar metal statues have masses in the ratio 1:8. They are made of the same material. Find the Linear Scale Factor.

Question 10[2 marks]Grade 7-8

Two similar cones have radii in the ratio 2:5. Find the ratio of their total surface areas.

Grade 9 Killer

Grade 9: Frustum via subtraction with fractional scaling

Question 11[4 marks]Grade 8-9

A cone has volume V. A smaller cone is removed from the top by cutting parallel to the base at 1/3 of the height from the apex. Find the volume of the remaining frustum in terms of V.

Question 12[3 marks]Grade 8-9

Two similar objects A and B are made of the same material (same density). If their masses are in the ratio 27:64, find the ratio of their surface areas.

Question 13[3 marks]Grade 8

The cost of painting a sculpture is proportional to its surface area. Two similar sculptures have Linear Scale Factor 2:1. If it costs £200 to paint the smaller sculpture, what does it cost to paint the larger one?

Question 14[3 marks]Grade 8-9

A large bottle holds 2 litres and has a height of 20 cm. A similar smaller bottle holds 500 ml. Find the height of the smaller bottle.

Question 15[3 marks]Grade 9

The surface area of sphere A is 4 times the surface area of sphere B. Find the ratio of their volumes, giving your answer in the form a:b.

Key Formula Summary (Higher Tier):

• Linear Scale Factor (LSF) = Length₂ ÷ Length₁

• Area Scale Factor (ASF) = LSF²

• Volume Scale Factor (VSF) = LSF³

• Reverse: LSF = √ASF or LSF = ³√VSF

• For same material: Mass ∝ Volume

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