GCSE Maths Geometry and Measure

Angles

Exterior Angles of a Polygon

# Exterior Angles of a Polygon

Here we will learn about exterior angles of polygons including how to calculate the sum of exterior angles for a polygon, a single exterior angle and how to use this knowledge to solve problems.

There are also angles in polygons worksheets based on Edexcel, AQA and OCR exam 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.

### What are exterior angles? ### 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 interior 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.
3. Solve the problem using the information you have already gathered.

### Explain how to solve problems involving exterior angles. ## 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.

2 Identify what the question is asking you to find.

The size of one exterior angle.

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

3 Solve the problem using the information you have already gathered.

$360 \div 6 = 60$

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?

8 sides and irregular – (irregular octagon).

As adjacent means next to we are being asked to find the size of the exterior angle which is on an straight line with the interior angle.

We know that angles on a straight line add to 180º, so if the interior angle is 130º then the exterior angle will be 50º.

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

Calculate angle x.

6 sides – irregular hexagon.

The interior angles of a hexagon add to 720 degrees.

Find the exterior angle x.

It is an irregular polygon so the exterior angles are not all equal.

We can work out the missing interior angle of the polygon.

\begin{aligned} 150+120+90+160+130+a&=720 \\\\ 650+a&=720 \\\\ a&=70 \end{aligned}

The interior angle  + the exterior angle must equal 180º

Therefore 70 + exterior angle = 180º

So x = 110º.

### 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?

Unknown number of sides.

Regular polygon – therefore all exterior angles are equal.

We need to find the number of sides.

We know the sum of the exterior angles is 360º and we know that each exterior angle is equal because it is a regular polygon.

\begin{aligned} 20 \times \text { number of sides }&=360\\\\ 20n&=360 \\\\ n&=18 \end{aligned}

Therefore the polygon has 18 sides.

### 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?

Unknown number of sides.

Regular polygon – therefore all exterior angles are equal.

We need to find the number of sides.

We know the sum of the exterior angles is 360º and we know that each exterior angle is equal because it is a regular polygon.

We also know that the sum of an interior and an exterior angle is 180º.

If the interior angle is 150º then the exterior angle will be 30º.

The number of sides can therefore be calculated by 360 ÷ 30 = 12

The polygon has 12 sides.

### 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.

Unknown sides.

Regular polygon – therefore each exterior angle is equal.

Number of sides of the polygon.

Other Information we know:

Total of Exterior Angles = 360º

Interior + Exterior Angle = 180º

11 × Interior Angle = Exterior Angle

We will call each of the interior angles x.

Since ‘11 × Interior Angle = Exterior Angle’ we can call each exterior angle 11x.

Therefore

\begin{aligned} x+11 x&=180 \\\\ 12 x&=180 \\\\ x&=15 \end{aligned}

The size of one exterior angle is 15º.

The number of sides of the polygon is 360 ÷ 15 = 24

The Polygon has 24 sides.

### 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

## The next lessons are

• Angles in parallel lines
• Congruent shapes
• Pythagoras’ theorem

## Still stuck?

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