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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

Year:  GCSE

GCSE Subjects:   Maths

GCSE Boards:   OCR, AQA, Eduqas, Pearson Edexcel,

Curriculum topic:   Basic Geometry, Geometry and Measures

Curriculum subtopic:   Angles Properties and Constructions

Popular topics:   Geometry worksheets

Difficulty level:  

Worksheet Overview

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.

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