Learning Objectives
- identify similar figures;
- use linear scale factors;
- calculate area and volume scale factors;
- use map and drawing scales;
- solve missing-length, area and volume problems.
Introduction
This lesson develops Similarity and Scale from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
Similarity
Similar shapes have the same shape but may have different sizes. Corresponding lengths are in the same ratio.
Clean Scale Diagram
Worked Example: Lengths
Two similar shapes have scale factor 3. A side on the smaller shape is 7 cm.
Corresponding larger side = 7×3 = 21 cm.
Worked Example: Areas
Linear scale factor is 4. Smaller area is 18 cm².
Area factor = 4²=16, so larger area = 18×16=288 cm².
Worked Example: Volumes
Linear scale factor is 2.5. Smaller volume is 32 cm³.
Volume factor=2.5³=15.625, so larger volume=500 cm³.
Scale Drawings
A scale such as 1:50 means 1 unit on the drawing represents 50 of the same units in reality.
A wall measures 8 cm on a 1:50 plan.
Real length=8×50=400 cm=4 m.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Linear scale factor is 5. A smaller side is 6 cm. Find the larger side.
- Linear factor is 3. Find the area factor.
- Linear factor is 4. Find the volume factor.
- Two similar figures have areas 25 cm² and 225 cm². Find the linear scale factor.
- Two similar solids have volume ratio 64:125. Find the linear ratio.
- A plan uses scale 1:200. A room measures 4.5 cm by 3 cm on the plan. Find actual dimensions in metres.
- Two similar cones have radii 4 cm and 10 cm. The smaller volume is 96π cm³. Find the larger volume.
✅ Check Your Work
Show practice answers
- 30 cm.
- 9.
- 64.
- Area factor=9, so linear factor=3.
- Linear ratio=4:5.
- 4.5×200=900 cm=9 m; 3×200=600 cm=6 m.
- Linear factor=10/4=2.5; volume factor=15.625; larger volume=1500π cm³.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Similar shapes have equal corresponding angles and proportional sides.
- Length factor k gives area factor k² and volume factor k³.
- Scale drawings must use consistent units.