# Similarity

> Edexcel International GCSE Mathematics A · 4MA1
> Source: https://www.owlsprep.com/study/edexcel-igcse-math-a-s4-similarity/

This guide covers all similarity content for Foundation and Higher tiers of Edexcel IGCSE Maths A (4MA1), including scale drawings, length ratios, and Higher-only area/volume ratio rules for similar shapes, with exam-focused worked examples.

**Prerequisites:** [Basic ratio calculations](https://www.owlsprep.com/study/edexcel-igcse-math-a-s1-ratios/); [Basic area and volume formulae](https://www.owlsprep.com/study/edexcel-igcse-math-a-s4-area-volume/)

## Learning objectives

- Recall that similar figures have equal corresponding angles and proportional corresponding side lengths
- Calculate unknown lengths using linear scale factors for maps, scale drawings and similar shapes (Foundation tier)
- Apply area ratio ($k^2$) and volume ratio ($k^3$) rules for similar figures (Higher tier)
- Solve multi-step exam problems involving similarity of 2D and 3D shapes

## Foundation: Similar Shape Properties & Scale Drawings

**Similar Figures** — Two figures are similar if all their corresponding angles are equal, and all corresponding side lengths are in the same ratio (called the linear scale factor $k$).

*Example:* Two equilateral triangles of side length 2cm and 6cm are similar, as all angles are 60° and the side length ratio is 1:3.

All scale drawings and maps are practical applications of similarity: the stated scale gives the linear scale factor between the drawing/map and the real object. For example, a scale of 1:1000 means 1cm on the map equals 1000cm (10m) in real life.

**Worked example:** A map has a scale of 1:20000. The distance between two towns on the map is 3.5cm. Calculate the real distance between the towns in kilometres.

1. Write the scale with units: 1cm on map = 20000cm real
2. $$3.5 \times 20000 = 70000 \text{ cm real}$$
3. Convert cm to km: 70000 cm = 700 m = 0.7 km
4. Final answer: 0.7 km

> **Exam tip:** Always explicitly write unit conversions for scale drawing questions to avoid losing easy marks.

## Foundation: Calculating Lengths of Similar Shapes

To find unknown lengths in similar shapes, first identify corresponding sides (sides opposite equal angles) to calculate the linear scale factor $k$. If scaling up from a smaller to larger shape, $k > 1$; if scaling down, $k < 1$.

> **tip**
>
> Label equal angles first to avoid mixing up corresponding sides, the most common mistake on length ratio questions.

**Worked example:** Triangles ABC and DEF are similar. AB = 4cm, DE = 12cm, BC = 3cm. Find the length of EF.

1. Identify corresponding sides: AB corresponds to DE, BC corresponds to EF
2. $$k = \frac{DE}{AB} = \frac{12}{4} = 3 \text{ (scaling from ABC to DEF)}$$
3. $$EF = 3 \times 3 = 9 \text{ cm}$$
4. Final answer: 9 cm

> **Exam tip:** Consistently define the direction of your scale factor (e.g. scaling from small to large shape) to avoid reciprocal errors.

## Higher: Area Ratio of Similar Shapes

**Area Scale Factor** — For similar 2D shapes, if the linear scale factor is $k$, the area scale factor is $k^2$. This applies to all similar shapes, including triangles, circles and composite figures.

*Example:* If linear scale factor is 3, area scale factor is $3^2 = 9$, so the larger shape has 9 times the area of the smaller shape.

**Worked example:** Two similar circles have radii in the ratio 2:5. The area of the smaller circle is 12 cm². Calculate the area of the larger circle.

1. $$k = \frac{5}{2} = 2.5 \text{ (linear scale factor from small to large)}$$
2. $$Area\ scale\ factor = k^2 = 2.5^2 = 6.25$$
3. $$Area\ of\ larger\ circle = 12 \times 6.25 = 75 \text{ cm}^2$$
4. Final answer: 75 cm²

> **Exam tip:** Never use linear scale factor directly for area calculations: you must square it first to get the correct area ratio.

*Calculator:* allowed

## Higher: Volume Ratio of Similar Shapes

**Volume Scale Factor** — For similar 3D solids, if the linear scale factor is $k$, the volume scale factor is $k^3$. This applies to all similar 3D shapes, including cubes, spheres, prisms and pyramids.

*Example:* If linear scale factor is 2, volume scale factor is $2^3 = 8$, so the larger solid has 8 times the volume of the smaller solid.

**Worked example:** Two similar cuboids have a linear scale factor of 1:4. The volume of the larger cuboid is 320 cm³. Find the volume of the smaller cuboid.

1. $$k = \frac{1}{4} \text{ (linear scale factor from large to small)}$$
2. $$Volume\ scale\ factor = k^3 = \left(\frac{1}{4}\right)^3 = \frac{1}{64}$$
3. $$Volume\ of\ smaller\ cuboid = 320 \times \frac{1}{64} = 5 \text{ cm}^3$$
4. Final answer: 5 cm³

> **Exam tip:** Double check you have cubed the linear scale factor for volume questions, as squaring it by mistake is a frequent error.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Mixing up corresponding sides when calculating scale factor
  - Why it fails: Corresponding sides must be opposite equal angles; using unmatched sides gives an incorrect scale factor
  - Correct: Mark equal angles first, then match sides opposite the same sized angles to form your ratio
- **Wrong:** Using linear scale factor directly for area or volume calculations
  - Why it fails: Area scales with the square of linear ratio, volume with the cube; using linear ratio directly gives the wrong result
  - Correct: Square linear scale factor for area questions, cube it for volume questions before applying it
- **Wrong:** Forgetting to convert units correctly in scale drawing questions
  - Why it fails: Map scales are usually in cm, while real distances are in m or km; unconverted units give answers that are orders of magnitude wrong
  - Correct: Write down the scale with units first, then convert your final answer to the required unit explicitly
- **Wrong:** Applying similarity rules to non-similar shapes
  - Why it fails: Shapes that look similar may not be: all corresponding angles must be equal for shapes to be similar
  - Correct: Only use similarity ratio rules if you are told the shapes are similar, or you can confirm all corresponding angles are equal
- **Wrong:** Using the wrong direction of scale factor (e.g. scaling down when you need to scale up)
  - Why it fails: If you use k from large to small when you need small to large, your answer will be the reciprocal of the correct value
  - Correct: State explicitly which shape you are scaling from and to before calculating k, e.g. 'k = large / small' if scaling up

## Cheatsheet

| Tier | Rule Type | Rule |
| --- | --- | --- |
| All Tiers | Similar Shapes Property | Equal corresponding angles, proportional sides |
| All Tiers | Linear Scale Factor | $k = \frac{length\ of\ shape\ 2}{length\ of\ shape\ 1}$ |
| All Tiers | Scale Drawings | 1 unit on drawing = k units real life |
| Higher Only | Area Scale Factor | $Area_{SF} = k^2$ |
| Higher Only | Volume Scale Factor | $Volume_{SF} = k^3$ |

## What's next

Now that you have mastered similarity rules, you can apply these to a wide range of geometry problems in your Edexcel IGCSE Maths A exam, including enlargement questions, composite shape problems, and multi-step 3D geometry questions. Similarity is often combined with trigonometry and circle theorems in higher tier exam questions, so make sure you revise those topics alongside this one to be prepared for the highest mark questions. Practice past paper questions on similarity to build speed and accuracy, paying close attention to unit conversions and scale factor direction to avoid common mistakes.

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