# Exterior Angles Of A Polygon

Here is everything you need to know about exterior angles in polygons for GCSE maths (Edexcel, AQA and OCR). You’ll learn how to calculate the sum of exterior angles for a polygon, a single exterior angle and use this knowledge to solve problems.

At the end you’ll find worksheets based on Edexcel, AQA and OCR GCSE exam style questions, along with further guidance on where to go next if you’re still stuck.

## What are exterior angles?

Exterior angles are angles between a polygon and the extended line from the vertex of the polygon.

Sum of exterior angles of a polygon = 360º

Interior and exterior angles form a straight line – they add to 180º.

Check out our lessons on interior angles of polygons and sum of the interior angles to find out more.

### Keywords

• Polygon:
A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines.

• Regular & irregular polygons:
A regular polygon is where all angles are equal size and all sides are equal length
E.g. a square.
An irregular polygon where all angles are not equal size and/or all sides are not equal length
E.g. a trapezium

## How to solve problems involving exterior angles

In order to solve problems involving exterior angles following these steps:

1. Identify the number of sides in any polygon/s given in the question.
Note whether these are regular or irregular polygons.
2. Identify what the question is asking recalling that the Sum of exterior angles \, \textbf{=} \; \bf{360^{\circ}}
3. Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of interior angles \, \textbf{=} \; \bf{(n-2) \times 180^{\circ}} if required

### Related lessons on angles in polygons

Exterior angles of a polygon is part of our series of lessons to support revision on angles in polygons. You may find it helpful to start with the main angles in polygons lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

## Exterior angles of polygons examples

### Example 1: finding the size of a single exterior angle for a regular polygon

Calculate the size of a single exterior angle for a regular hexagon.

1. Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular shapes.

A hexagon has 6 sides.

Regular – therefore all exterior angles are equal.

2Identify what the question is asking recalling that the Sum of exterior angles \, \textbf{=} \; \bf{360^{\circ}}

The size of one exterior angle.

We know the sum of exterior angles for a polygon is 360°.

3Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of interior angles \, \textbf{=} \; \bf{(n-2) \times 180^{\circ}} if required

$360 \div 6 = 60$

The size of each exterior angle is 60º.

### Example 2: finding an exterior angle given an interior angle for an irregular polygon

An irregular octagon has one interior angle of size 130º. What is the size of the adjacent exterior angle?

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons.

Identify what the question is asking recalling that the Sum of exterior angles \, \textbf{=} \; \bf{360^{\circ}}

Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of interior angles \, \textbf{=} \; \bf{(n-2) \times 180^{\circ}} if required

### Example 3: interior + exterior angle = 180º

Calculate angle x.

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons.

Identify what the question is asking recalling that the Sum of exterior angles \, \textbf{=} \; \bf{360^{\circ}}

Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of interior angles \, \textbf{=} \; \bf{(n-2) \times 180^{\circ}} if required

### Example 4: finding the number of sides given the exterior angle of a regular polygon

An exterior angle of a regular polygon is 20º. How many sides does the polygon have?

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons.

Identify what the question is asking recalling that the Sum of exterior angles \, \textbf{=} \; \bf{360^{\circ}}

Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of interior angles \, \textbf{=} \; \bf{(n-2) \times 180^{\circ}} if required

### Example 5: finding the number of sides given the interior angle of a regular polygon

The size of each interior angle of a regular polygon is 150º. How many sides does the polygon have?

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons.

Identify what the question is asking recalling that the Sum of exterior angles \, \textbf{=} \; \bf{360^{\circ}}

Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of interior angles \, \textbf{=} \; \bf{(n-2) \times 180^{\circ}} if required

### Example 6: multi step problem involving interior and exterior angles

The size of each interior angle of a regular polygon is 11 times the size of each exterior angle. Work out the number of sides the polygon has.

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons.

Identify what the question is asking recalling that the Sum of exterior angles \, \textbf{=} \; \bf{360^{\circ}}

Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of interior angles \, \textbf{=} \; \bf{(n-2) \times 180^{\circ}} if required

### Common misconceptions

• Misidentifying the exterior angle

E.g.

The exterior angle of a triangle is the angle between the side and the extension of an adjacent side.

Here the interior angle (internal angle) is 60º, so the exterior angle (external angle) must be 120º.

• Miscounting the number of sides
• Misidentifying if a polygon is regular or irregular
• Incorrectly assuming all the angles are the same size
• Misidentifying which angle the question is asking you to calculate

### Practice exterior angles of a polygon questions

1. Find the size of one exterior angle for a regular quadrilateral.

90^{\circ}

60^{\circ}

180^{\circ}

270^{\circ}

Exterior angles of a polygon add up to 360 . A regular quadrilateral has 4 interior angles equal in size, so the four exterior angles are equal.

This means we can divide 360 by 4 to get the solution.

2. Find the size of one exterior angle for a regular octagon.

45^{\circ}

60^{\circ}

40^{\circ}

135^{\circ}

Exterior angles of a polygon add up to 360 . A regular octagon has 8 interior angles equal in size, so the eight exterior angles are equal.

This means we can divide 360 by 8 to get the solution.

3. Find the size of one exterior angle for a regular nonagon.

90^{\circ}

40^{\circ}

140^{\circ}

280^{\circ}

Exterior angles of a polygon add up to 360 . A regular nonagon has 9 interior angles equal in size, so the nine exterior angles are equal.

This means we can divide 360 by 9 to get the solution.

4. Each of the exterior angles of a regular polygon is 12^{\circ} .

How many sides does the polygon have?

 12  sides

 20  sides

 30  sides

 32  sides

Exterior angles of a polygon add up to 360 .

This means we can divide 360 by 12 to get the solution.

5. Each of the exterior angles of a regular polygon is 20^{\circ} .

How many sides does the polygon have?

 12  sides

 20  sides

 18  sides

 16  sides

Exterior angles of a polygon add up to 360 .

This means we can divide 360 by 20 to get the solution.

6. Four interior angles in a pentagon are  125^{\circ} each.

Find the size of the other angle

 125^{\circ}

 40^{\circ}

 55^{\circ}

 140^{\circ}

The four known exterior angles will be 55^{\circ} , since angles on a straight line sum to 180 . This means the fifth exterior angle will be 140^{\circ} because exterior angles add up to 360 .

Using angles on a straight line once more means that the missing angle is 40^{\circ} .

### Exterior angles of a polygon GCSE questions

1.  A regular polygon has 15 sides. Calculate the size of one exterior angle.

(1 mark)

360 ÷ 15 = 24

= 24^{\circ}

(1)

2.  (a)  The diagram below shows part of a regular polygon. Calculate the size of the exterior angle of the polygon.

(b)  Work out how many sides this polygon has.

(3 marks)

(a)

180 – 162=18

= 18^{\circ}

(1)

(b)

360 ÷ 18

(1)

20

(1)

3.   Shown below are parts of two regular polygons.

Polygon A has 9 sides and an exterior angle of x.

Polygon B has an interior angle of 3x.

How many sides does polygon B have?

(4 marks)

360 ÷ 9 = 40

(1)

x = 40,   3x = 120

(1)

Polygon B: Interior angle is  120^{\circ} , exterior angle is 60^{\circ}

(1)

360 \div 60 = 6

(1)

## Learning checklist

You have now learned how to:

• Use conventional terms for geometry e.g. exterior angle
• Derive a formula for the total of exterior angles for a polygon and consequently calculate the sum of exterior angles for a regular polygon
• Solve problems involving interior and exterior angles

## Still stuck?

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