Polygons

Q: What is the sum of exterior angles of any polygon?

The sum of exterior angles of any polygon, regardless of the number of sides, is always 360°. For regular polygons, you can divide this by (number of sides) to find the size of each equal exterior angle.

Q: Do I need to memorise the polygon angle sum formula for the exam?

Yes, the formula for the sum of interior angles of an -sided polygon, , is not provided on the formula sheet, so you must recall it.

Edexcel International GCSE Mathematics A· 4.2· 18 min read

1. Identifying Polygons and Special Quadrilaterals★★☆☆☆⏱ 4 min

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📘 Definition

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°

📘 Definition

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
1. Parallelograms have opposite equal sides, so this is not a match
3
1. 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.

2. Sum of Interior Angles of Polygons★★★☆☆⏱ 5 min

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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
1. 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.

3. Regular Polygons: Interior and Exterior Angles★★★☆☆⏱ 5 min

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📘 Definition

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:
1. The sum of exterior angles of any polygon (regular or irregular) is always 360°, so each exterior angle of a regular $n$ -gon = $\frac{36 0 ^{\circ}}{n}$
2. Each interior angle = 180° minus the exterior angle at the same vertex, or $\frac{( n - 2 ) \times 18 0 ^{\circ}}{n}$ .

📐 Worked Example

Calculate the size of each interior angle of a regular octagon.

1
1. An octagon has 8 sides, so $n = 8$
2
1. 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
1. 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.

4. Congruence of Polygons★★☆☆☆⏱ 3 min

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📘 Definition

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
1. 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.

5. Common Pitfalls

Wrong move:

Assuming trapeziums have two pairs of parallel sides

Why:

Only parallelograms have two pairs of parallel sides; trapeziums are defined as having exactly one pair of parallel sides

Correct move:

Quote exactly one pair of parallel sides when defining or identifying a trapezium

Wrong move:

Using the wrong number of sides $n$ for named polygons

Why:

Mixing up polygon names (e.g. hexagon = 6 sides, not 5) leads to incorrect angle sum calculations

Correct move:

Memorise side counts for common polygons: pentagon=5, hexagon=6, octagon=8

Wrong move:

Thinking the sum of exterior angles changes with number of sides

Why:

Students often assume more sides = larger exterior angle sum, but it is always fixed at 360° for any polygon

Correct move:

Recall that exterior angle sum is 360°, regardless of the number of sides of the polygon

Wrong move:

Claiming a square is not a type of rhombus

Why:

Students incorrectly assume rhombuses can only have non-right angles, but squares meet all rhombus definition criteria

Correct move:

Remember squares are a special case of both rhombuses and rectangles

Wrong move:

Using 180° sum for quadrilaterals instead of 360°

Why:

Mixing up triangle and quadrilateral interior angle sums leads to missing angle errors

Correct move:

Memorise quadrilateral angle sum is 360°, or derive it using the general formula for $n = 4$

6. Quick Reference Cheatsheet

Concept	Rule/Formula
Sum of interior angles (n sides)	$(n - 2) \times 18 0^{\circ}$
Sum of exterior angles (any polygon)	$36 0^{\circ}$
Regular n-gon exterior angle	$\frac{36 0 ^{\circ}}{n}$
Regular n-gon interior angle	$18 0^{\circ} - \frac{36 0 ^{\circ}}{n}$
Quadrilateral interior angle sum	$36 0^{\circ}$
Congruent polygons	Same shape and size, all corresponding sides/angles equal

7. Frequently Asked

What is the sum of exterior angles of any polygon?

The sum of exterior angles of any polygon, regardless of the number of sides, is always 360°. For regular polygons, you can divide this by $n$ (number of sides) to find the size of each equal exterior angle.

Is a square a type of rhombus?

Yes. A square meets all the defining properties of a rhombus (four equal sides, opposite sides parallel, opposite angles equal) plus the extra property of all internal angles being 90°.

Do I need to memorise the polygon angle sum formula for the exam?

Yes, the formula for the sum of interior angles of an $n$ -sided polygon, $(n - 2) \times 18 0^{\circ}$ , is not provided on the formula sheet, so you must recall it.

Going deeper

official_resourceEdexcel IGCSE Maths A SpecificationFull official specification for Geometry section 4

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.