Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Similar Shapes & Scale Factors

In one sentence. When one shape is an enlargement of another: lengths scale by k, areas by k², volumes by k³ — and how to run those factors in any direction.

Mensuration — one of 4 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Galileo Galilei. Learn Similar Shapes & Scale Factors, teach it, then defend it against Galileo Galilei.

Where this sits in the specification

Specification 4MA1, 4.11 Similarity (area and volume factors: Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 4.11 — Similarity (area and volume factors: Higher)

1 · Understand it

What it is

Similar shapes have the same shape, different size: every length in one is k times the matching length in the other. Areas then scale by k² and volumes by k³. Given any pair of matching measurements, k (or k² or k³) can be found — and converted to the others by squaring, cubing or rooting.

Why it works

Area is length × length — scale each length by k and the area picks up k twice: k². Volume is length³, so it collects k³. This is why a model boat 1/10 the length holds a thousandth of the cargo, and why giants can't exist at human proportions — weight grows as k³ while bone strength (a cross-sectional AREA) grows only as k².

When a question is asking for it

Matching sides of similar triangles; 'the areas are 25 cm² and 64 cm²…' (k² given — square-root to reach k); model/map/real problems; and any mixed question hopping between lengths, areas and volumes.

How to hold it in your head

One ladder with three rungs: LENGTH ×k, AREA ×k², VOLUME ×k³. Whatever pair of facts the question gives, climb to the rung it lives on, then move up or down the ladder by rooting or powering. Never move sideways by guessing.

2 · The mistakes that cost marks

  • Scaling area by k: doubling lengths QUADRUPLES area. The carpet for a doubled room costs 4×, not 2×.
  • Scaling volume by k or k²: doubling lengths multiplies volume by 8.
  • Given two areas, using their ratio as k. The area ratio IS k² — square-root it to get the length factor.
  • Matching non-corresponding sides in similar triangles. Match by ANGLE position, not by appearance.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

In triangle ABC, D is on AB and E is on AC, with DE parallel to BC. AD = 4 cm, DB = 6 cm and DE = 5 cm. Work out BC.

  1. Triangles ADE and ABC are similar: they share angle A, and angle ADE = angle ABC (corresponding angles, DE ∥ BC).
  2. Match corresponding sides by position: AD matches AB — the WHOLE side, AB = 4 + 6 = 10 cm, not DB.
  3. k = AB/AD = 10/4 = 2.5.
  4. BC = 5 × 2.5 = 12.5 cm.

Answer: 12.5 cm

Two similar cones have volumes 54 cm³ and 250 cm³. The smaller cone has a surface area of 36 cm². Work out the surface area of the larger cone.

  1. The given ratio is on the VOLUME rung: k³ = 250/54 = 125/27.
  2. Climb down to lengths: k = ∛(125/27) = 5/3.
  3. Climb up to areas: k² = 25/9.
  4. Surface area: 36 × 25/9 = 100 cm².

Answer: 100 cm²

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Two similar triangles have corresponding sides of 6 cm and 9 cm. A second side of the smaller triangle is 8 cm. Work out the corresponding side of the larger triangle. (answer in cm)
  2. Grade 7 — Two similar shapes have corresponding sides of 4 cm and 10 cm. A second side of the smaller shape is 5 cm. Work out the corresponding side of the larger shape. (answer in cm)
  3. Grade 8 — Two similar shapes have a length scale factor of 3. The smaller shape has area 12 cm². Work out the area of the larger shape. (answer in cm²)
  4. Grade 9 — Two similar bottles have a length scale factor of 2. The smaller bottle holds 50 cl. How much does the larger bottle hold? (answer in cl)
  5. 9+ — Two similar solids have surface areas of 36 cm² and 81 cm². The smaller solid has a volume of 160 cm³. Work out the volume of the larger solid. (answer in cm³)

Ask the person who made it

Inside Incandio you can put a question to Leonardo da Vinci, who answers knowing exactly which module you have just finished: “Your notebooks obsess over proportion — the Vitruvian Man is one giant exercise in similar figures and scale. Why did you believe proportion was the key to drawing, building and understanding the body?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Similar Shapes & Scale Factors, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Similar Shapes & Scale Factors works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Similar Shapes & Scale Factors against Galileo Galilei — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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