Mathematics 4024 · Syllabus tracker
Similarity
Lengths in similar shapes, showing triangles are similar, nested triangles, and area and volume scale factors.
How to use this
Read each "I can…" statement and rate yourself honestly. Then try its quick check. If you get a check wrong on something you rated Got it, we'll suggest moving it to Shaky. My progress compares your ratings with your past-paper results to find topics you think you know but keep losing marks on.
Loading the tracker…
Quick revision
Similarity on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
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
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.