This module covers shapes that are the same shape but not always the same size (similar) and shapes that are identical in both (congruent). You learn how to match corresponding sides, use a scale factor for lengths, and see how areas and volumes change with it. Map scales and scale models are the everyday versions.
It matters because the same small set of ideas turns up in geometry proofs, scale drawings, maps, enlargements and problem-solving questions. Check the current syllabus on the Cambridge page to confirm which parts apply to your tier and version.
What should you know first?
You should be comfortable with ratios and fractions, and with finding missing angles in triangles. If either feels rusty, look at ratio and proportional reasoning and angles and geometric reasoning first.
An orienting example
Triangle X has sides 6, 8 and 10 cm. Triangle Y is similar, with sides 9, 12 and 15 cm.
Linear scale factor. 9 ÷ 6 = 1.5, and 12 ÷ 8 = 1.5, and 15 ÷ 10 = 1.5. Every length is multiplied by 1.5.
Area. Triangle X is right-angled, because 6² + 8² = 100 = 10². Its area is ½ × 6 × 8 = 24 cm². Triangle Y has area ½ × 9 × 12 = 54 cm². The ratio is 54 ÷ 24 = 2.25, which is 1.5².
Volume. If these were the cross-sections of two similar prisms, the volumes would be in the ratio 1.5³ = 3.375.
One scale factor, three different effects: k for lengths, k² for areas, k³ for volumes. The five lessons below take this apart.
In what order should you study the lessons?
- Identify corresponding sides reliably. Every later step depends on pairing the right sides, especially when a diagram is turned or overlapping.
- Use a linear scale factor. Finding and applying the factor is the core skill, including for maps and models.
- Relate area change to a scale factor. Area uses the factor squared, and this is the most commonly confused step.
- Relate volume change to a scale factor. Volume uses the factor cubed and is usually combined with capacity or mass.
- Distinguish similarity from congruence. This gives you the vocabulary and conditions for proof-style questions.
Then test everything together in the mixed practice set.
What traps catch students in this topic?
- Pairing by letter order instead of by equal angles.
- Adding or subtracting a length difference instead of multiplying by a factor.
- Using k instead of k² for area, or k² instead of k³ for volume.
- Inverting the factor, so an enlargement gets smaller or a reduction gets larger. A quick sense check catches it.
- Forgetting unit conversions when a map scale or a capacity is involved.
- Claiming congruence from information that only proves similarity, or from an angle that is not between the two given sides.
How should you use the practice set?
Work through the mixed practice after lessons 1 to 3 at least, and again after lesson 5. Write full working, then compare with the solution. Keep missed questions in the mistake log and retest them a few days later.
The map scale, contour and gradient practice tool gives extra scale questions. If one lesson keeps returning wrong answers, an experienced teacher in online one-to-one Mathematics tuition can focus your time there.