Use the length scale factor k to get area scale factor k2 and volume scale factor k3 for similar shapes and solids. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
If lengths scale by k, areas scale by k2 and volumes (and mass, at constant density) scale by k3. A length scale factor of 3 makes area ×9 and volume ×27. The reverse: if areas are in the ratio 4:9, lengths are 2:3 (square-root), and volumes are 8:27.
Write which factor you are using. Mixing them — multiplying an area by k instead of k2 — is the whole error this topic exists to catch. Similar solids must be actually similar (all lengths in the same ratio), not just 'both cylinders'.
Length factor k, area k2, volume k3. Going backwards: cube-root a volume ratio to get lengths. Do not apply the length factor to an area.
Objectives
Square a length scale factor to get the area scale factor.
Cube a length scale factor to get the volume scale factor.
Go back from an area or volume scale to the length scale.
Key points
Find the length scale factor from a pair of corresponding lengths.
Square it for areas (including surface areas) and cube it for volumes.
Work backwards: square-root an area factor or cube-root a volume factor to get back to lengths.
Move between area and volume by going through the length factor.
Worked examples
Example 1
Length scale factor 3. What is the area scale factor?
2 marks
Identify that area scale factor is the square of the length scale factor: k^2.
Calculate the square: 3^2 = 9.
The area scale factor is 9.
Answer: 9
Common slips
Slip: Leave scale factor as 3. Area scales with the square of linear dimensions.
Slip: Dan doubled 3 to get 6. Area requires squaring, not doubling.
Slip: Conclude 3. The area scale factor is 9.
Example 2
Length scale factor 2. What is the volume scale factor?
2 marks
Identify that volume scale factor is the cube of the length scale factor: k^3.
Calculate the cube: 2^3 = 8.
The volume scale factor is 8.
Answer: 8
Common slips
Slip: Square the length scale factor to get 4. Squaring gives the area scale factor, but volume requires cubing.
Slip: Eve wrote 4. Volume scales with three dimensions: 2 × 2 × 2 = 8.
Slip: Conclude 6. 2 cubed is 8, not 2 × 3.
Example 3
Area scale factor 25. What is the length scale factor?
3 marks
Identify that to reverse from area scale to length scale, take the square root.
Calculate the square root: √25 = 5.
The length scale factor is 5.
Answer: 5
Common slips
Slip: Divide 25 by 2. Area scale factor is k^2, so length scale factor is the square root of 25.
Slip: Flo divided by 2 to get 12.5. Go back with a square root, not halving.
Slip: Conclude 25. The linear scale factor is 5.
Questions and answers
GCSE Maths similar shapes: area & volume questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Length scale factor 4. What is the area scale factor?
Show answer
Answer: 16
Area scale is k^2.
4^2 = 16.
M1 — Area scale is k^2.
A1 — 4^2 = 16.
Question 2 Easy2 marks
Length scale factor 5. What is the area scale factor?
Show answer
Answer: 25
Area scale is k^2.
5^2 = 25.
M1 — Area scale is k^2.
A1 — 5^2 = 25.
Question 3 Medium2 marks
Length scale factor 3. What is the volume scale factor?
Show answer
Answer: 27
Volume scale is k^3.
3^3 = 27.
M1 — Volume scale is k^3.
A1 — 3^3 = 27.
Question 4 Medium2 marks
Length scale factor 4. What is the volume scale factor?
Show answer
Answer: 64
Volume scale is k^3.
4^3 = 64.
M1 — Volume scale is k^3.
A1 — 4^3 = 64.
Question 5 Hard2 marks
Area scale factor 16. What is the length scale factor?
Show answer
Answer: 4
Length scale is √k.
√16 = 4.
M1 — Length scale is √k.
A1 — √16 = 4.
Question 6 Hard2 marks
Area scale factor 36. What is the length scale factor?
Show answer
Answer: 6
Length scale is √k.
√36 = 6.
M1 — Length scale is √k.
A1 — √36 = 6.
Question 7 Exam-style4 marks
Two similar solids have volumes 8 cm³ and 64 cm³. The smaller solid has surface area 12 cm². Work out the surface area of the larger solid.
Show answer
Answer: 48
The volume scale factor is 64 ÷ 8 = 8.
The length scale factor is the cube root of 8, which is 2.
The area scale factor is 2^2 = 4.
The larger surface area is 12 × 4 = 48 cm².
M1 — Volume scale factor 64 ÷ 8 = 8
A1 — Length scale factor 2
M1 — Squares the length scale factor: 4
A1 — 48
Common mistakes
Most lost marks on this topic are one slip.
Using the length scale unchanged for an area
Square the length scale
Squaring when the question asks for volume
Cube the length scale
Halving an area scale to get back to length
Take the square root
Spot the slip
A model has length scale factor 3. Cyd squares nothing and says the area scale factor is 3. What is it?
9
3
6
Show the slip
9 Area scale is the square of the length scale.
Exam tips
A calculator is useful here. Show the calculation you typed, not only the final line.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the three factors as a short table: k, k2, k3 — the marker can then follow every step.
Cube roots of 8, 27, 64, 125 and square roots of 4, 9, 16, 25 appear constantly; know them instantly.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.