# Polygons

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

This guide covers all polygon content tested in Edexcel IGCSE Mathematics A (4MA1), including naming shapes, quadrilateral properties, interior/exterior angle calculations, and congruence rules for polygons.

**Prerequisites:** [Basic angle rules (straight line, angles around a point)](https://www.owlsprep.com/study/edexcel-igcse-math-a-s4-basic-angle-rules/); [Properties of triangles](https://www.owlsprep.com/study/edexcel-igcse-math-a-s4-triangles/)

## Learning objectives

- Name and identify common polygons and special quadrilaterals
- Calculate the sum of interior angles of any polygon using the $(n-2) \times 180^\circ$ formula
- Find interior and exterior angles of regular polygons
- Apply properties of special quadrilaterals to solve angle and side problems
- Define congruence for polygons and identify congruent shapes

## Identifying Polygons and Special Quadrilaterals

**Polygon** — A closed 2D shape with 3 or more straight, non-intersecting sides, no curved edges.

*Example:* Triangle (3 sides), pentagon (5 sides), octagon (8 sides)

- Named polygons you must recognise: pentagon (5 sides), hexagon (6 sides), octagon (8 sides)
- All 4-sided polygons are called quadrilaterals, with interior angles summing to 360°

**Quadrilateral** — A polygon with exactly 4 sides, sum of internal angles = 360°

| Quadrilateral | Key Defining Properties |
| --- | --- |
| Parallelogram | Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other |
| Rectangle | Parallelogram with all angles 90°; diagonals equal in length |
| Square | Rectangle with all sides equal; diagonals perpendicular, bisect vertex angles |
| Rhombus | Parallelogram with all sides equal; diagonals perpendicular, bisect vertex angles |
| Trapezium | Exactly one pair of parallel opposite sides |
| Kite | Two pairs of adjacent equal sides; one pair of opposite equal angles |

**Worked example:** Name the quadrilateral with two pairs of adjacent equal sides and one pair of opposite equal angles.

1. 1. Compare the given properties to the quadrilateral definitions
2. 2. Parallelograms have *opposite* equal sides, so this is not a match
3. 3. Kites are defined as having two pairs of adjacent equal sides and one pair of opposite equal angles
4. Final answer: Kite

> **Exam tip:** When asked to name a quadrilateral in the exam, always quote 2-3 key properties to justify your answer and get full marks.

*Calculator:* allowed

## Sum of Interior Angles of Polygons

The sum of interior angles of any $n$-sided polygon can be calculated using the formula below, derived from splitting the polygon into $n-2$ triangles, each with an interior angle sum of 180°.

$$Sum\ of\ interior\ angles = (n - 2) \times 180^\circ = (2n - 4) \times 90^\circ$$

**Worked example:** Calculate the sum of interior angles of a 7-sided heptagon.

1. 1. Identify the number of sides: $n=7$
2. 2. Substitute into the formula:
3. $$Sum = (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$$
4. Final answer: 900°

**Worked example:** A quadrilateral has angles of 72°, 105° and 121°. Find the size of the fourth angle.

1. 1. Sum of interior angles of a quadrilateral is 360°
2. $$2. Sum of given angles = 72 + 105 + 121 = 298^\circ$$
3. $$3. Missing angle = 360 - 298 = 62^\circ$$
4. Final answer: 62°

> **Exam tip:** For quadrilateral angle questions, you can use either the general $n=4$ formula or recall the fixed 360° sum to save time.

*Calculator:* allowed

## Regular Polygons: Interior and Exterior Angles

**Regular Polygon** — A polygon where all sides are equal length and all interior angles are equal size.

*Example:* Square, regular hexagon, regular octagon

Two key rules for regular polygons: <br>1. The sum of exterior angles of *any* polygon (regular or irregular) is always 360°, so each exterior angle of a regular $n$-gon = $\frac{360^\circ}{n}$ <br>2. Each interior angle = 180° minus the exterior angle at the same vertex, or $\frac{(n-2) \times 180^\circ}{n}$.

**Worked example:** Calculate the size of each interior angle of a regular octagon.

1. 1. An octagon has 8 sides, so $n=8$
2. 2. Use the faster exterior angle method:
3. $$Exterior\ angle = \frac{360^\circ}{8} = 45^\circ$$
4. $$Interior\ angle = 180^\circ - 45^\circ = 135^\circ$$
5. Final answer: 135°

**Worked example:** A regular polygon has an interior angle of 150°. How many sides does it have?

1. 1. First calculate the exterior angle at the same vertex:
2. $$Exterior\ angle = 180^\circ - 150^\circ = 30^\circ$$
3. 2. Rearrange the exterior angle formula to solve for n:
4. $$n = \frac{360^\circ}{exterior\ angle} = \frac{360}{30} = 12$$
5. Final answer: 12 sides (dodecagon)

> **Exam tip:** Always use the exterior angle formula for questions where you are given an interior angle and asked for the number of sides, as it requires fewer steps and reduces calculation errors.

*Calculator:* allowed

## Congruence of Polygons

**Congruent Polygons** — Polygons that are exactly the same shape and size, with all corresponding sides and angles equal. Rotation or reflection of a shape does not change congruence.

*Example:* Two squares with side length 5cm are congruent; a square of side 3cm and a square of side 6cm are not congruent.

**Worked example:** State if two equilateral triangles with side lengths 7cm and 8cm are congruent, giving a reason.

1. 1. Check for matching shape and size:
2. 2. Both are equilateral triangles, so they are the same shape, but their side lengths are different so they are different sizes.
3. Final answer: No, their corresponding sides are not equal lengths.

> **Exam tip:** When explaining congruence in the exam, you must reference both matching shape and matching size (or equal corresponding sides/angles) to get full marks.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Assuming trapeziums have two pairs of parallel sides
  - Why it fails: Only parallelograms have two pairs of parallel sides; trapeziums are defined as having exactly one pair of parallel sides
  - Correct: Quote exactly one pair of parallel sides when defining or identifying a trapezium
- **Wrong:** Using the wrong number of sides $n$ for named polygons
  - Why it fails: Mixing up polygon names (e.g. hexagon = 6 sides, not 5) leads to incorrect angle sum calculations
  - Correct: Memorise side counts for common polygons: pentagon=5, hexagon=6, octagon=8
- **Wrong:** Thinking the sum of exterior angles changes with number of sides
  - Why it fails: Students often assume more sides = larger exterior angle sum, but it is always fixed at 360° for any polygon
  - Correct: Recall that exterior angle sum is 360°, regardless of the number of sides of the polygon
- **Wrong:** Claiming a square is not a type of rhombus
  - Why it fails: Students incorrectly assume rhombuses can only have non-right angles, but squares meet all rhombus definition criteria
  - Correct: Remember squares are a special case of both rhombuses and rectangles
- **Wrong:** Using 180° sum for quadrilaterals instead of 360°
  - Why it fails: Mixing up triangle and quadrilateral interior angle sums leads to missing angle errors
  - Correct: Memorise quadrilateral angle sum is 360°, or derive it using the general formula for $n=4$

## Cheatsheet

| Concept | Rule/Formula |
| --- | --- |
| Sum of interior angles (n sides) | $(n-2) \times 180^\circ$ |
| Sum of exterior angles (any polygon) | $360^\circ$ |
| Regular n-gon exterior angle | $\frac{360^\circ}{n}$ |
| Regular n-gon interior angle | $180^\circ - \frac{360^\circ}{n}$ |
| Quadrilateral interior angle sum | $360^\circ$ |
| Congruent polygons | Same shape and size, all corresponding sides/angles equal |

## What's next

Now that you have mastered polygon properties and angle calculations, you are ready to move on to more advanced geometry topics in the Edexcel IGCSE Mathematics A specification. Next, you will apply these angle rules to problems involving circles, congruent triangles, and trigonometry, which make up a large portion of the geometry and trigonometry unit. You should also practice applying polygon properties to multi-step geometry problems, as these are frequently tested in both foundation and higher tier papers. Make sure you memorise all the rules and formulae in this guide, as none are provided on the exam formula sheet. Regular practice of past paper questions will help you avoid common errors and build speed when solving angle problems.

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