# Similarity & Symmetry

> Mathematics · CIE IGCSE 0580
> Source: https://www.owlsprep.com/study/cie-0580-u4-similarity-symmetry/

This guide covers all Core and Extended content for Similarity & Symmetry in CIE IGCSE Maths 0580, including similarity length calculations, 2D symmetry, and Extended-only scale factor ratios and 3D symmetry rules.

**Prerequisites:** [Basic 2D shape properties](https://www.owlsprep.com/study/cie-0580-u4-2d-shapes/); [Foundational mensuration skills](https://www.owlsprep.com/study/cie-0580-u4-mensuration/)

## Learning objectives

- Calculate missing side lengths of similar 2D shapes for Core tier
- Identify line and rotational symmetry in 2D figures
- Formally prove triangle similarity for Extended tier
- Apply $k:k^2:k^3$ length:area:volume ratios for similar solids (Extended)
- Recognise symmetry properties of common 3D solids (Extended)

## Similar Shapes (Core Tier)

**Similar Shapes** — Shapes that have identical corresponding angles and proportional corresponding side lengths. You can scale, rotate or reflect one similar shape to exactly match the other.

*Example:* Two equilateral triangles of different sizes are similar

The ratio of corresponding side lengths of two similar shapes is called the **linear scale factor $k$**. To find a missing side length, multiply the corresponding known side length on the original shape by $k$.

**Worked example:** Two similar rectangles have corresponding side lengths of 4 cm and 12 cm. The height of the smaller rectangle is 3 cm. Find the height of the larger rectangle.

1. Calculate linear scale factor from small to large: $k = \frac{12}{4} = 3$
2. Multiply the smaller height by $k$: $3 \times 3 = 9$ cm
3. Final answer: 9 cm

> **Exam tip:** Always confirm scale factor direction first: small → large = $k>1$, large → small = $k<1$ to avoid inverse errors.

## 2D Symmetry (Core Tier)

**Line Symmetry** — A shape has line symmetry if there exists a line (axis of symmetry) that divides the shape into two identical mirror-image halves.

**Rotational Symmetry** — A shape has rotational symmetry if it maps onto itself after rotation of less than 360° around its center. The number of times it maps onto itself in one full rotation is its order of rotational symmetry.

- Square: 4 lines of symmetry, order 4 rotational symmetry
- Isosceles triangle: 1 line of symmetry, order 1 rotational symmetry (no rotational symmetry)
- Parallelogram: 0 lines of symmetry, order 2 rotational symmetry

**Worked example:** State the number of lines of symmetry and order of rotational symmetry of a regular hexagon.

1. A regular hexagon has 6 equal sides and angles, so it has 6 lines of symmetry (each running from a vertex to the midpoint of the opposite side)
2. It maps onto itself every $60°$ ($\frac{360}{6} = 60$), so order of rotational symmetry is 6

> **Exam tip:** Order 1 rotational symmetry means the shape only matches itself after a full 360° rotation, so it counts as having no rotational symmetry for exam purposes.

## Extended Addition 1: Similarity Proof & Scale Factor Ratios

**Triangle Similarity Proof Conditions** — Two triangles are similar if any one of the following holds: 1. AA (all corresponding angles equal), 2. SSS (all corresponding sides in equal ratio), 3. SAS (two corresponding sides in equal ratio, included angle equal).

**Worked example:** Prove triangle PQR is similar to triangle XYZ, given $\angle P = \angle X = 65°$, $\frac{PQ}{XY} = \frac{PR}{XZ} = 2.5$.

1. We have two pairs of corresponding sides in equal ratio: $\frac{PQ}{XY} = \frac{PR}{XZ} = 2.5$
2. The included angle between these sides is equal for both triangles: $\angle QPR = \angle YXZ = 65°$
3. By SAS similarity condition, triangles PQR and XYZ are similar

For all similar figures, if the linear scale factor is $k$, then:
- Area scale factor = $k^2$
- Volume scale factor = $k^3$

**Worked example:** Two similar spheres have linear scale factor 2. The smaller sphere has surface area $16\pi$ cm². Find the surface area of the larger sphere.

1. Area scale factor = $k^2 = 2^2 = 4$
2. Multiply small surface area by area scale factor: $16\pi \times 4 = 64\pi$ cm²
3. Final answer: $64\pi$ cm²

> **Exam tip:** Never mix linear, area and volume scale factors: square $k$ for area, cube $k$ for volume before applying to the original value.

## Extended Addition 2: 3D Symmetry

Extended students must recognise symmetry properties of common 3D solids, including planes of symmetry (3D equivalent of lines of symmetry) and axes of rotational symmetry.

| Solid | Planes of Symmetry | Rotational Symmetry Axis (order) |
| --- | --- | --- |
| Cube | 9 | 3 axes (order 4), 4 axes (order 3), 6 axes (order 2) |
| Right circular cylinder | Infinite + 1 perpendicular to axis | 1 axis (infinite order) |
| Square-based pyramid | 4 | 1 axis (order 4) |
| Regular triangular prism | 4 | 1 axis (order 3) |

**Worked example:** State the number of planes of symmetry of a square-based pyramid.

1. A square-based pyramid has a square cross-section at its base, with 4 lines of symmetry
2. Each line of symmetry of the base corresponds to a plane of symmetry cutting through the pyramid apex and the base line of symmetry
3. Total planes of symmetry = 4

> **Exam tip:** For prisms, count planes of symmetry the same way you count lines of symmetry for the cross-section, then add 1 perpendicular plane halfway along the prism length if applicable.

## Common pitfalls

- **Wrong:** Using linear scale factor for area/volume calculations
  - Why it fails: Area and volume scale with the square and cube of the linear factor respectively, not linearly
  - Correct: Square $k$ for area calculations, cube $k$ for volume calculations before multiplying by the original value
- **Wrong:** Counting order 1 rotational symmetry as having rotational symmetry
  - Why it fails: Order 1 means the shape only matches itself after a full 360° rotation, which is true for all shapes so it does not count as rotational symmetry
  - Correct: Only state a shape has rotational symmetry if its order is 2 or higher
- **Wrong:** Using inverse scale factor (e.g. small→large $k$ for large→small calculations)
  - Why it fails: This produces an incorrectly sized length/area/volume, leading to lost marks
  - Correct: Test your scale factor first: large→small = $k<1$, small→large = $k>1$
- **Wrong:** Counting 2 lines of symmetry for a standard parallelogram
  - Why it fails: Reflecting a parallelogram over its diagonal or midline does not produce a mirror image
  - Correct: Only special parallelograms (rectangle, rhombus, square) have lines of symmetry
- **Wrong:** Forgetting to confirm corresponding angles are equal for similarity proof
  - Why it fails: Proportional sides alone do not prove similarity unless the included angle is equal (SAS) or all sides are proportional (SSS)
  - Correct: Use only the AA, SSS or SAS formal conditions to prove triangle similarity

## Cheatsheet

| Concept | Core Tier Rule | Extended Tier Add-On |
| --- | --- | --- |
| Similar shapes | Missing side = corresponding side × linear scale factor $k$ | Prove similarity via AA/SSS/SAS; Area = $k^2$ × original area, Volume = $k^3$ × original volume |
| 2D Symmetry | Count lines of symmetry; count order of rotational symmetry | Count planes of symmetry and rotational axes for 3D solids |
| Scale factor direction | $k>1$ = small→large, $k<1$ = large→small | Same direction rule applies for area/volume scale factors |

## What's next

Now that you have mastered Similarity & Symmetry for CIE IGCSE Maths 0580, you can apply these concepts to more advanced geometry topics. Similarity ratios are particularly useful for mensuration questions involving frustums and composite solids, while symmetry rules help you solve problems involving tessellations and coordinate geometry. Practice past paper questions to reinforce your understanding of both Core and Extended content, paying special attention to scale factor ratio questions which are high-value common Extended Paper 4 questions.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-0580-u4-similarity-symmetry/