Square and cube the length factor for similar areas and volumes
Use k² for areas and surface areas · Use k³ for volumes of similar solids · Recover a length factor from an area or volume ratio
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Similar lengths
- Square and cube roots
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use k² for areas and surface areas
- Use k³ for volumes of similar solids
- Recover a length factor from an area or volume ratio
Dimensions determine the power
A length factor k affects two independent directions of an area, so area and surface-area factors are k². It affects three directions of a volume, so the volume factor is k³. These conclusions require geometrically similar figures or solids, not merely one enlarged dimension.
The pictured length factor 1.5 gives area factor 2.25 and, for similar solids, volume factor 3.375. Angle measures remain unchanged. Keep the scale direction consistent for every ratio.
Reverse the relationship
For a positive area ratio A₂/A₁, the length factor is √(A₂/A₁). For a positive volume ratio V₂/V₁, it is ∛(V₂/V₁). Choose positive roots for length scales; state the corresponding direction and units.
Scaling dimensions does not by itself predict mass unless an additional density assumption is supplied. These tasks ask only for geometry.
Sources for this lesson
Pause and explain
For length factor 3, what is the volume factor?
Worked example
Similar solids have larger-to-smaller volume ratio 27. Find their length and surface-area ratios.
Show the worked solution
- Length factor = ∛27 = 3.
- Surface-area ratio = 3² = 9.
Answer Length ratio 3; surface-area ratio 9.
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.
Similar plane figures have length factor 2 and smaller area 15 cm². Find the larger area.
Give me a hint
Square the length factor.
Compare my reasoning
- Area factor = 4.
- 15×4 = 60 cm².
60 cm².
Look for these in your work
- Use two dimensions
- Keep area units
Similar plane figures have larger-to-smaller area ratio 81:25. Find their length ratio.
Give me a hint
Take the positive square root of the area ratio.
Compare my reasoning
- k = √(81/25) = 9/5.
- The length ratio is 9:5.
9:5, in the same larger-to-smaller direction.
Look for these in your work
- Reverse the squared relationship
- Use a positive length factor
A cube has surface area 2400 cm² and volume 8000 cm³. A similar model uses length scale 1:20. Find its surface area and volume.
Give me a hint
Use (1/20)² and (1/20)³.
Compare my reasoning
- Surface area = 2400/400 = 6 cm².
- Volume = 8000/8000 = 1 cm³.
Surface area 6 cm²; volume 1 cm³.
Look for these in your work
- Keep both dimensional powers
- Use model-to-original direction
Common mistakes
- Do not multiply area or volume by k alone.
- Similarity and matching scale direction are required.
What can you explain now?
For length factor 3, what is the volume factor?
Compare with the explanation
27.
Three scaled dimensions contribute 3³.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
For length factor 3, what is the volume factor?
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.