Quick revision notes · Mathematics 0580 · Chapter 30 of 37
Similarity
Similar shapes, scale factors for lengths, areas and volumes, and nested triangles.
3 sectionsWorked examples
Common mistakes
At a glance
Similar shapes, scale factors for lengths, areas and volumes, and nested triangles.
1Similar shapes and lengths
- Similar shapes have equal angles and sides in the same ratio.
- Length scale factor k = a length on the new shape ÷ the matching length on the old one.
- Two triangles are similar if two pairs of angles are equal (the third then matches too).
How to do it
- Identify matching sides (opposite equal angles).
- Find k from a pair of known matching sides.
- Multiply (or divide) to find the missing length.
Worked example
Two similar triangles have matching sides 4 cm and 10 cm. A side of 6 cm on the small one matches which length?
- k = 10 ÷ 4 = 2.5.
- 6 × 2.5.
15 cm
2Nested triangles
- A line parallel to one side of a triangle cuts off a smaller similar triangle.
- Use the whole sides of the big triangle, not just the extra piece.
- Redraw the two triangles separately the same way round.
Worked example
In triangle ABC, DE ∥ BC with D on AB, E on AC. AD = 3, DB = 6, DE = 4. Find BC.
- AB = 3 + 6 = 9.
- k = 9 ÷ 3 = 3.
- BC = 4 × 3.
12
3Areas and volumes of similar shapes
- Length factor k → area factor k2 → volume factor k3.
- Surface area scales like area (k2); mass and capacity scale like volume (k3).
- Work backwards with roots: area ratio → k = √(area ratio); volume ratio → k = ∛(volume ratio).
Formulas
area factor = k2
volume factor = k3
k = √(area ratio) = ∛(volume ratio)
Worked example
Two similar jugs have surface areas 180 cm² and 405 cm². The small one holds 0.8 litres. How much does the large one hold?
- Area factor = 405 ÷ 180 = 2.25, so k = 1.5.
- Volume factor = 1.53 = 3.375.
- 0.8 × 3.375.
2.7 litres
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Area doubles when the lengths double | Area × 4 (k2 = 22) |
| Using DB instead of AB for the big triangle | Use the whole side AB |
| Cubing the volume ratio to get a length | Take the cube root |
| Showing similarity with one pair of equal angles | You need two pairs of equal angles |
★In the exam
- Write k clearly as a fraction or decimal before using it.
- For area or volume questions, find the length factor first.
- To prove similarity, state the equal angles with reasons.