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In order to access this I need to be confident with:

2D polygons

Angles on a straight line

Finding missing angles in a triangle

Geometrical properties of triangles, squares, rectangles, parallelograms, kites and trapeziums

Geometrical properties of regular vs irregular polygons

This topic is relevant for:

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.

**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.*

**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

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

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

Get your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREECalculate the size of a single exterior angle for a regular hexagon.

**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

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

\[360 \div 6 = 60\]

The size of each exterior angle is

An irregular octagon has one interior angle of size

**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 you to find.**

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.

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

We know that angles on a straight line add to

Calculate angle

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

The interior angles of a hexagon add to 720 degrees.

**Identify what the question is asking you to find.**

Find the exterior angle

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

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

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

Therefore

So

An exterior angle of a regular polygon is

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

Unknown number of sides.

Regular polygon – therefore all exterior angles are equal.

**Identify what the question is asking you to find.**

We need to find the number of sides.

We know the sum of the exterior angles is

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

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

Therefore the polygon has

The size of each interior angle of a regular polygon is

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

Unknown number of sides.

Regular polygon – therefore all exterior angles are equal.

**Identify what the question is asking you to find.**

We need to find the number of sides.

We know the sum of the exterior angles is

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

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

If the interior angle is

The number of sides can therefore be calculated by

The polygon has

The size of each interior angle of a regular polygon is

Unknown sides.

Regular polygon – therefore each exterior angle is equal.

**Identify what the question is asking you to find.**

Number of sides of the polygon.

Other Information we know:

Total of Exterior Angles ** **360º

Interior ** **180º

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

We will call each of the interior angles

Since

Therefore

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

The size of one exterior angle is

The number of sides of the polygon is

The Polygon has

**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 ^{º}

**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**

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:

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} .

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

**(1 mark)**

Show answer

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)**

Show answer

(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)**

Show answer

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)**

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

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