Similar areas and volumes
Square or cube the length factor, and reverse those relationships to recover a length ratio.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Use corresponding length ratios.
- Calculate squares, cubes and roots.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 1.5²?
What you will learn
- Use k² for areas and surface areas and k³ for volumes.
- Recover a length factor from an area or volume ratio.
- Explain when these rules require genuinely similar shapes.
Each dimension brings another factor
If every length is multiplied by k, an area involves two length dimensions and is multiplied by k². A 4 by 3 rectangle enlarged by k = 1.5 becomes 6 by 4.5: its area changes from 12 to 27, a factor of 2.25. Surface areas of similar solids also use k² because every face area scales that way.
Volumes involve three dimensions, so similar solids use k³. Doubling every dimension gives four times the surface area but eight times the volume. These rules need uniform scaling: doubling only a box's height while keeping its base fixed does not create a similar box, and its volume only doubles.
Work backwards with roots
An area ratio 9 : 25 gives a length ratio 3 : 5, using the positive square roots. A volume ratio 8 : 27 gives a length ratio 2 : 3, using cube roots. Once you have the length factor, use it to find a missing side or raise it to another power for the requested measure.
For two similar containers with volumes 64 and 216 cm³, the larger-over-smaller volume factor is 216/64 = 27/8. The length factor is 3/2 and the surface area factor is 9/4. Keep the ratio direction consistent throughout. Length units, square units and cubic units cannot be interchanged even when their numerical values look similar.
Pause and explain
Similar solids have volume ratio 8 : 27. What is their length ratio?
Worked example
A similar solid is enlarged by length factor 1.5. Its original surface area is 40 cm² and volume is 80 cm³. Find the new measures.
Show the worked solution
- Surface area factor is 1.5² = 2.25; new area is 40 × 2.25 = 90 cm².
- Volume factor is 1.5³ = 3.375; new volume is 80 × 3.375 = 270 cm³.
- State the different units and check both factors came from the same length multiplier.
Answer 90 cm² and 270 cm³.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A similar shape has length factor 3 and original area 14 cm². Find its new area.
Give me a hint
Square the length factor before multiplying the area.
Compare my reasoning
- The area factor is 3² = 9.
- New area is 14 × 9 = 126 cm².
- Check that the multiplier nine, not three, reflects two scaled dimensions.
126 cm².
Look for these in your work
- Squared the length factor.
- Used square units.
Two similar solids have length ratio 2 : 5. The smaller volume is 24 cm³. Find the larger volume.
Give me a hint
Use larger-over-smaller length factor 5/2.
Compare my reasoning
- The volume multiplier is (5/2)³ = 125/8.
- Multiply 24 × 125/8 = 375.
- The larger volume is 375 cm³; reverse check gives ratio 24:375 = 8:125.
375 cm³.
Look for these in your work
- Cubed the ratio in the correct direction.
- Retained cubic units.
Similar models have surface areas 72 cm² and 200 cm². The smaller height is 9 cm and volume 54 cm³. Find the larger height and volume.
Give me a hint
Recover the length factor from the area ratio first.
Compare my reasoning
- Area factor is 200/72 = 25/9, so length factor is 5/3.
- Height becomes 9 × 5/3 = 15 cm.
- Volume becomes 54 × (5/3)³ = 250 cm³.
Height 15 cm; volume 250 cm³.
Look for these in your work
- Used a square root to recover length scaling.
- Then used the cube of that length factor for volume.
Common mistakes
- Multiplying an area or volume by the length factor alone.
- Taking a square root of a volume ratio.
- Applying similarity rules when only one dimension changes.
What can you explain now?
Similar shapes have area ratio 16 : 81. What is their length ratio?
Compare with the explanation
4 : 9
Take positive square roots of both area-ratio terms to obtain corresponding length ratios.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, choose a length factor and explain its effects on perimeter, surface area and volume without consulting the formulas.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.