Use similarity and scale factors
Match corresponding sides, scale lengths and relate length, area and volume factors for similar figures and solids.
Section 5 · Geometry
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What stays equal in similar shapes?
What you will learn
- Calculate lengths in similar shapes
- Explain triangle similarity from angles or side ratios
- Use squared and cubed factors for area and volume
Similarity preserves shape, not size
Similar figures have equal corresponding angles and proportional corresponding lengths. A scale factor k maps each length to its matching length.
For triangles, state the correspondence before comparing sides. Similarity may be justified by equal angles or matching side ratios; proving congruence is not required.
Area and volume use powers of the scale factor
If lengths scale by k, areas and surface areas scale by k² and volumes by k³. Keep the direction of the ratio consistent.
Core focuses on lengths of similar shapes; Extended includes area and volume relationships for similar solids and geometric explanations of triangle similarity.
Pause and explain
If all lengths of a solid double, how does its volume change?
Worked example
A model side is 4 cm and the corresponding real side is 10 cm. Find the length and area scale factors from model to real object.
Show the worked solution
- Length factor = 10/4 = 2.5.
- Area factor = 2.5² = 6.25.
- The real area is 6.25 times the model area.
Answer Length factor 2.5; area factor 6.25.
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 triangles have corresponding sides 5 cm and 8 cm. Find the scale factor and the matching length to a 7 cm side.
Give me a hint
Use the same correspondence and direction in each ratio.
Compare my reasoning
- Factor from first to second = 8/5 = 1.6.
- Multiply the matching 7 cm side by 1.6.
- The new length is 11.2 cm.
Scale factor 1.6; length 11.2 cm.
Look for these in your work
- Sides correspond correctly
- Ratio direction consistent
- Length scaled once
Similar solids have length factor 3 from small to large. If the small volume is 12 cm³, find the large volume and the surface-area factor.
Give me a hint
Use k³ for volume and k² for area.
Compare my reasoning
- Volume factor = 3³ = 27.
- Large volume = 12 × 27 = 324 cm³.
- Surface-area factor = 3² = 9.
Volume 324 cm³; surface-area factor 9.
Look for these in your work
- Scale direction clear
- Volume exponent 3 used
- Area exponent 2 used
An architectural model uses a 1:40 length ratio. A room is 30 cm² on the model. Find the real area in m².
Give me a hint
Square the length ratio, then convert square centimetres to square metres.
Compare my reasoning
- Real-to-model length factor = 40, so area factor = 40² = 1600.
- Real area = 30 × 1600 = 48 000 cm².
- Since 10 000 cm² = 1 m², the area is 4.8 m².
4.8 m².
Look for these in your work
- Ratio direction identified
- Area factor squared
- Units converted
Common mistakes
- Comparing non-corresponding sides
- Using k for area instead of k²
- Using k² for volume instead of k³
- Switching ratio direction halfway through
What can you explain now?
If corresponding lengths have scale factor 3, what is the area factor?
Compare with the explanation
9
Square the length factor.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why do lengths, areas and volumes use k, k² and k³ for similar figures?
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.