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Calculate Angle Sizes Within Polygons

In this worksheet, students will calculate angle sizes within polygons. They will also find the number of sides of a polygon when given an interior angle.

'Calculate Angle Sizes Within Polygons ' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   OCR, AQA, Eduqas, Pearson Edexcel

Curriculum topic:   Basic Geometry, Geometry and Measures

Curriculum subtopic:   Angles, Properties and Constructions

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Polygon is the name given to any 2D shape with straight sides.

A regular polygon has all sides and angles which are the same, whilst an irregular polygon has different side lengths and angles.

 

Read on to explore some key facts relating to polygons and their interior angles

 

Diagram of interior angles

 

This table summarises the information you hopefully already know about polygons and their interior angles:

Table of polygons

We know that a triangle has angles that add up to 180°.

 

Did you know that we can use this fact to help us find the sum of the angles in any polygon?

 

Diagram showing angles in polygons

As you can see in the diagram above, we can split a quadrilateral into two triangles.

 

Each triangle has three angles that add up to 180°, therefore 180° + 180° = 360°.

 

 

Triangles in a pentagon digram

 

Here, we can split a pentagon into three triangles.

Therefore, 3 × 180° = 540°

 

 

Can you spot the pattern?

 

4 sides = 2 triangles × 180°

5 sides = 3 triangles × 180°

 

Can you see that the number of triangles in a polygon is always two less than its number of sides?

 

 

We can make a formula from this:

Sum of the degrees in a polygon = (number of sides - 2) × 180

 

Therefore, for a hexagon (6-sided shape), we don't need to draw the shape to find the angle sum, we can just work it out:

6 - 2 = 4

4 × 180° = 720°

 

 

So our final summary table will look like this:

Angles in polygons table

If we know the sum of the angles in a regular polygon, how do we find the size of a single angle?

 

Cartoon boy with question marks

 

All we need to do is take the sum of all the angles, then divide this by the number of sides present.

e.g. The interior angle of a pentagon is:

540° ÷ 5 = 108°

 

 

And finally, if we are given an interior angle, we can find the number of sides of the shape, like this:

Interior angle = 150°

180 - 150 = 30

360 ÷ 30 = 12 sides

 

 

 

Now let's put all these facts and formulae into practice in some real questions now. 

 

In this activity, we will calculate angle sizes within polygons. We will also find the number of sides of a polygon when given an interior angle.

Review the regular polygon below:

 

Green heptagon

A 7-sided shape is called a heptagon.

 

Find the sum of the interior angles, then select the correct answers to complete the expressions below. 

 1260°900°1440°128.5°205.7°178°
The sum of the angles is...
An interior angle is...

Next, consider the regular polygon below:

 

Orange decagon

 

Find the sum of the interior angles, then select the correct answers to complete the expressions below. 

 1440°1260°1800°144°126°180°
The sum of the interior angles is...
One interior angle is...

This is an irregular octagon.

 

Diagram of an irregular octagon with labelled angles

What is the size of the missing angle labelled x?

 1440°1260°1800°144°126°180°
The sum of the interior angles is...
One interior angle is...

Boy peeping over a book

 

What is the size of a single interior angle within a regular polygon with 20 sides?

 1440°1260°1800°144°126°180°
The sum of the interior angles is...
One interior angle is...

Here is an irregular heptagon:

Diagram showing known and unknown angles using algebra

What is the value of x in this diagram?

62°

48°

53°

59°

Little girl thinking

 

"Can you help me fill in the blank with a number, please?"

62°

48°

53°

59°

Review the known and unknown angles on the diagram below, some of which have been expressed algebraically:

Diagram showing known and unknown angles using algebra

What is the value of x in this diagram?

The value of x is 40° / 50° / 60°.

Cartoon image of a girl asking questions

 

Find the number of sides of a polygon with interior angles totalling 1260°.

6 sides

9 sides

8 sides

5 sides

Let's try another...

 

Find the number of sides of a polygon with interior angles totalling 8640°.

6 sides

9 sides

8 sides

5 sides

Cartoon of boy confused

 

It's match up time!

 

Match each interior angle on the left to its corresponding polygon.

 9 sides10 sides15 sides20 sides30 sides40 sides
Interior angle of 140°
Interior angle of 162°
Interior angle of 171°
  • Question 1

Review the regular polygon below:

 

Green heptagon

A 7-sided shape is called a heptagon.

 

Find the sum of the interior angles, then select the correct answers to complete the expressions below. 

CORRECT ANSWER
 1260°900°1440°128.5°205.7°178°
The sum of the angles is...
An interior angle is...
EDDIE SAYS
Did you draw this shape and divide it into triangles or use the formula? Either way, it doesn't matter which method we choose - both should give the correct answer. A heptagon has 7 sides, and could be split into 5 triangles. 5 × 180° = 900° Similarly, if we use our formula, we will reach the same answer: Sum of the angles = (number of sides - 2) × 180 Sum = (7 - 2) × 180 = 900° To calculate a single interior angle, we need to divide this total by the number of interior angles: 900 ÷ 7 = 128.5°
  • Question 2

Next, consider the regular polygon below:

 

Orange decagon

 

Find the sum of the interior angles, then select the correct answers to complete the expressions below. 

CORRECT ANSWER
 1440°1260°1800°144°126°180°
The sum of the interior angles is...
One interior angle is...
EDDIE SAYS
Did you remember the formula? Sum of the angles = (number of sides - 2) × 180 10 sides - 2 = 8 triangles 8 × 180 = 1440° To find one interior angle, we need to divide the total by the number of interior angles present: 1440 ÷ 10 = 144°
  • Question 3

This is an irregular octagon.

 

Diagram of an irregular octagon with labelled angles

What is the size of the missing angle labelled x?

CORRECT ANSWER
EDDIE SAYS
The difference here is that we are working with an irregular polygon. To calculate the sum of the angles, the principal remains the same: Sum of the angles = (number of sides - 2) × 180 8 - 2 = 6 6 × 180 = 1080° As this is an irregular octagon, we know that each of the angles will not be the same. Therefore, we have to add up the angles given and subtract these from 1080° to find our missing angle: 150 + 160 + 140 + 100 + 170 + 120 + 110 = 950° 1080 - 950 = 130°
  • Question 4

Boy peeping over a book

 

What is the size of a single interior angle within a regular polygon with 20 sides?

CORRECT ANSWER
EDDIE SAYS
We hope you are not hiding behind your book! This time, we want only the size of one of the interior angles. We can only calculate this in this scenario because we are working with a regular polygon. First, let's find the total angle sum for the whole shape: Sum of the angles = (number of sides - 2) × 180 20 sides - 2 = 18 triangles 18 × 180 = 3240° As we only want to find the size of one of the interior angles, all we need to do is take the sum of all the angles, then divide this by the number of sides present: 3240 ÷ 20 = 162°
  • Question 5

Here is an irregular heptagon:

Diagram showing known and unknown angles using algebra

What is the value of x in this diagram?

CORRECT ANSWER
59°
EDDIE SAYS
Here we go again, but we have got this! First, let's find the sum of the angles: Sum of the angles = (number of sides - 2) × 180 7 - 2 = 5 5 × 180° = 900° Now let's create our expression using the information provided: 2x + (2x + 10) + 2x + (2x - 5) + (2x + 30) + 135° + 140° = 900° Then, we can simplify: 10x + 310 = 900 10x = 590 x = 590 ÷ 10 x = 59°
  • Question 6

Little girl thinking

 

"Can you help me fill in the blank with a number, please?"

CORRECT ANSWER
EDDIE SAYS
Who would have thought that the humble triangle could be so useful to us? Remember to take 2 away from the number of sides of the polygon we are working with, then multiply by 180°: 9 sides - 2 = 7 triangles 7 × 180 = 1260° Were you able to help this girl out?
  • Question 7

Review the known and unknown angles on the diagram below, some of which have been expressed algebraically:

Diagram showing known and unknown angles using algebra

What is the value of x in this diagram?

CORRECT ANSWER
The value of x is 40° / 50° / 60°.
EDDIE SAYS
Whoa, what's going on here? We have some algebra included here, and we need to use two sets of skills together. A four sides polygon has angles which add up to 360°. Now, let's substitute the values we know into an expression: 2x + (x + 20) + (2x + 5) + 85° = 360° If we simplify the algebra, we reach: 5x + 110 = 360 5x = 250 x = 50° How did you get on there?
  • Question 8

Cartoon image of a girl asking questions

 

Find the number of sides of a polygon with interior angles totalling 1260°.

CORRECT ANSWER
9 sides
EDDIE SAYS
Just when you thought it was plain sailing... a curve ball! All we have to do here is work backwards. We know the formula to find the sum of interior angles is: Sum = (number of sides - 2) × 180 To find the number of sides when we have been given the sum of the internal angles, we need to work backwards through this formula: 1260 = (N - 2) × 180 1260 ÷ 180 = N - 2 7 = N - 2 7 + 2 = N Number of sides = 9
  • Question 9

Let's try another...

 

Find the number of sides of a polygon with interior angles totalling 8640°.

CORRECT ANSWER
EDDIE SAYS
Formulas are great, aren't they? Once we know them, we can always play around with them to help us find out what we need to know. We know the formula to find the sum of interior angles is: Sum = (number of sides - 2) × 180 If we have been given the sum of the angles, we can find the missing number of sides (N) instead: 8640 = (N - 2) × 180 8640 ÷ 180 = N - 2 48 = N - 2 48 + 2 = N N = 50
  • Question 10

Cartoon of boy confused

 

It's match up time!

 

Match each interior angle on the left to its corresponding polygon.

CORRECT ANSWER
 9 sides10 sides15 sides20 sides30 sides40 sides
Interior angle of 140°
Interior angle of 162°
Interior angle of 171°
EDDIE SAYS
Did you remember the rule to apply in this scenario? Take the given angle away from 180° then divide 360 by this answer. 180 - 140 = 40 360 ÷ 40 = 9 sides 180 - 162 = 18 360 ÷ 18 = 20 sides 180 - 171 = 9 360 ÷ 9 = 40 sides Great job! You can now calculate angle sizes within polygons or find the number of sides of a polygon when given an interior angle.
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