The smart way to improve grades

Comprehensive & curriculum aligned

Try an activity or get started for free

Calculate External Angles of a Polygon

In this worksheet, students will calculate the exterior or external angles of polygons with a variety of different sides by applying the rule that all exterior angles in a polygon will add up to 360°.

'Calculate External Angles of a Polygon' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Angles

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Student with phone

 

Lost again?

 

Not to worry the GPS on your phone can help you find your way.

 

GPS uses angles, including exterior angles, to help calculate and create directions.

The mathematics used in GPS is quite complex, but we can learn about some of the basics to get us thinking a bit more like a GPS system. 

 

 

Let's start by considering what we need to know about exterior angles around points and polygons. 

 

 

Diagram showing exterior angles example

 

 

exterior angles example

 

 

Because the exterior angles of a polygon will fit around a point, and we know that angles around a point add up to 360°, we just use this fact, along with the number of sides of the polygon, to calculate any exterior angle. So:

360° ÷ number of sides = value of external angle

 

Hopefully, now you won't get lost!

 

 

 

In this activity, we will calculate the exterior or external angles of polygons with a variety of different sides by applying the rule that all exterior angles in a polygon will add up to 360°. 

Review the regular polygon below:

 

A purple hexagon

 

Calculate the size of the exterior angle of this hexagon.

55°

40°

60°

65°

Next, consider the regular polygon below:

 

A blue nonagon (9-sided polygon)

 

Now type a number to complete the sentence below. 

55°

40°

60°

65°

Explore this regular polygon:

 

A regular, purple decagon

 

Calculate the exterior angle of this shape and type a number to complete the sentence below. 

55°

40°

60°

65°

Young boy on a bike

 

"Can you help me?

 

I need to find the exterior angle of a polygon.

 

How do I do this?"

Divide the number of sides by 360°

Multiply the number of sides by 360°

Divide 360° by the number of sides

Cartoon of a child looking confused

 

What is the exterior angle of a 20-sided shape?

Divide the number of sides by 360°

Multiply the number of sides by 360°

Divide 360° by the number of sides

Photo of a thoughtful young girl

 

If a shape has an exterior angle of 24°, how many sides does it have?

It's match up time!

 

Match each single exterior angle on the left to its corresponding polygon on the right.

Column A

Column B

10°
36 sides
72 sides
60 sides
15°
24 sides

Review the known and unknown angles on the diagram below:

Diagram of a triangle with known and unknown angles

What is the value of angle x?

Column A

Column B

10°
36 sides
72 sides
60 sides
15°
24 sides

Shape A is a 12-sided shape with an exterior angle of 3x.

 

The exterior angle of shape B is 2x.

Diagram showing external angle algebra

Then select the correct answers to complete the expressions below. 

 36°12°10°12161518
The value of x is...
The number of sides shape B has is...

Photo of a young girl thinking

 

What is the exterior angle of an 11-sided shape?

 

Round your answer to 1 decimal place.

 36°12°10°12161518
The value of x is...
The number of sides shape B has is...
  • Question 1

Review the regular polygon below:

 

A purple hexagon

 

Calculate the size of the exterior angle of this hexagon.

CORRECT ANSWER
60°
EDDIE SAYS
A hexagon has six sides so all we need to do is divide 360° by 6: 360 ÷ 6 = 60° Pretty straightforward, right? Let's try applying this formula to some more complex shapes now.
  • Question 2

Next, consider the regular polygon below:

 

A blue nonagon (9-sided polygon)

 

Now type a number to complete the sentence below. 

CORRECT ANSWER
EDDIE SAYS
This shape is a nonagon, which is the official name of a 9-sided shape. Did you know that one? The exterior angle of this nonagon can be calculated by dividing 360° by its number of sides: 360 ÷ 9 = 40° Hopefully, this sum is becoming ingrained in your brain now!
  • Question 3

Explore this regular polygon:

 

A regular, purple decagon

 

Calculate the exterior angle of this shape and type a number to complete the sentence below. 

CORRECT ANSWER
EDDIE SAYS
Okay, so we weren't given the number of sides this time... but that's not a problem, we just need to count them carefully. This is a decagon or a 10-sided shape. So to calculate the exterior angle, we need to work out: 360 ÷ 10 = 36° Are you navigating these activities well so far?
  • Question 4

Young boy on a bike

 

"Can you help me?

 

I need to find the exterior angle of a polygon.

 

How do I do this?"

CORRECT ANSWER
Divide 360° by the number of sides
EDDIE SAYS
Did you recall this key fact from the Introduction? As all the exterior angles of a polygon fit around a point, we can simply divide 360° by the number of sides to calculate the value of each individual external angle. We need to make sure we get this calculation the correct way around, otherwise we will get some weird and wonderful answers!
  • Question 5

Cartoon of a child looking confused

 

What is the exterior angle of a 20-sided shape?

CORRECT ANSWER
EDDIE SAYS
Bring it on! It doesn't matter how large the number of sides gets, does it? 360 ÷ 20 = 18°
  • Question 6

Photo of a thoughtful young girl

 

If a shape has an exterior angle of 24°, how many sides does it have?

CORRECT ANSWER
15
EDDIE SAYS
This time we need to work backwards. We know how to find the exterior angle of a polygon by dividing 360° by the number of sides present. So to find the number of sides, we need to divide 360° by the exterior angle given then hey presto! 360 ÷ 24 = 15 sides
  • Question 7

It's match up time!

 

Match each single exterior angle on the left to its corresponding polygon on the right.

CORRECT ANSWER

Column A

Column B

10°
36 sides
60 sides
72 sides
15°
24 sides
EDDIE SAYS
Whether we are given external angles or the number of sides, it's no bother - what a great little formula to know! We simply need to apply one of these two formulae: 360 ÷ number of sides = exterior angle 360 ÷ exterior angle given = number of sides Let's look at this in action using one pair as an example: 360 ÷ 10° = 36 sides 360 ÷ 36 sides = 10° So 10° and 36 sides are our first matching pair, can you find the others? Which way round did you choose to work?
  • Question 8

Review the known and unknown angles on the diagram below:

Diagram of a triangle with known and unknown angles

What is the value of angle x?

CORRECT ANSWER
EDDIE SAYS
Sometimes a question can be presented in a way which can make things look more complicated than they are. A triangle is still a type of polygon - it is just a specific type with three sides. So we can calculate the external angle (x) like this: 360 ÷ 3 = 120° Can you see the connection here? The external angle of a triangle is the same as the sum of the two interior opposite angles, which is another key angle fact to remember.
  • Question 9

Shape A is a 12-sided shape with an exterior angle of 3x.

 

The exterior angle of shape B is 2x.

Diagram showing external angle algebra

Then select the correct answers to complete the expressions below. 

CORRECT ANSWER
 36°12°10°12161518
The value of x is...
The number of sides shape B has is...
EDDIE SAYS
So this time we have some algebra thrown in too - don't worry! Just apply what we know as usual. Let's find the exterior angle of shape A first: 360 ÷ 12 = 30° This angle represents '3x' so to find x, we can divide this by 3: 30° ÷ 3 = 10 x = 10 We can now apply this value for x, to find the value of the exterior angle of shape B: 2 × 10 = 20° We know that we can divide 360° by the value of an external angle to find the number of sides present: 360 ÷ 20 = 18 sides Amazing! You can now calculate the exterior or external angles of polygons with a variety of different sides by applying the rule that all exterior angles in a polygon will add up to 360°.
  • Question 10

Photo of a young girl thinking

 

What is the exterior angle of an 11-sided shape?

 

Round your answer to 1 decimal place.

CORRECT ANSWER
EDDIE SAYS
Did you know that the name for an 11-sided shape is a hendecagon? A strange name, don't you think? We can calculate the size of the exterior angle using: 360 ÷ 11 = 32.7°
---- OR ----

Get started for free so you can track and measure your child's progress on this activity.

What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started
laptop

Try an activity or get started for free