 # 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. 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:   ### QUESTION 1 of 10 My interior design company is called 999.  The police thought it was an inside job.

Before I get to do the interior design, architects and builders have been there first. They love angles as it helps them a great deal in their work. They would need to know how to find interior angles of various shapes.

A polygon is any shape with straight sides. A regular polygon has all sides and angles the same, an irregular polygon has different side lengths and angles.

Lets see how this is done.  We know that a triangle has angles that add up to 180°

Did you know that we can use this to help us find the sum of the angles in any polygon? As you can see we can split a quadrilateral into two triangles. Each triangle has angles that add up to 180°, therefore 180° + 180° = 360°  Here you can see that we can split a pentagon into 3 triangles.  Therefore, 3 x 180° = 540°  Can you spot the pattern?

4 sides = 2 triangles x 180°

5 sides = 3 triangles x 180°

Can you see that the number of triangles is two less than the number of sides?

We can make a formula from this.

The sum of the degrees in a polygon =  number of sides - 2 x 180

Therefore for a hexagon (6 sided shape) we don't need to draw the shape.

6 - 2 = 4.  4 x 180° = 720° If we know the sum of the angles in a polygon, how do we find the size of one angle?

All you do is take the sum of all the angles ÷ by the number of sides.

Example

The interior angle of a pentagon is 540° ÷ 5 = 108°

And finally If you are given an interior angle you can find the number of sides like this

Interior angle = 150°

180 - 150 = 30

360 ÷ 30 = 12 sides A seven sided shape is called a heptagon.

Find the sum of the interior angles.

Find the sum of one angle.

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

Find the value of one of the interior angles.

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

What is the size of the missing angle. What is the size of an interior angle with 20 sides? Here is an irregular heptagon.

What is the value of x?

62°

48°

53°

59° Can you help me fill in the blanks please.

62°

48°

53°

59° What is the value of x?

The value of x is 40°, 50° 60° Find the number of sides of a polygon with interior angles adding up to 1260&deg;

6

9

8

5 Lets try another

Find the number of sides of a polygon with interior angles adding up to 8640° Find the number of sides for the following.

 9 10 15 20 30 40 Interior angle 140° Interior angle 162° Interior angle 171°
• Question 1 A seven sided shape is called a heptagon.

Find the sum of the interior angles.

Find the sum of one angle.

 1260° 900° 1440° 128.5° 205.7° 178° The sum of the angles is An interior angle is
EDDIE SAYS
Did you draw the shape, or use the formula? Either way it doesn't matter which method you use. A heptagon has 7 sides, which would give us 5 triangles. 5 x 180° = 900° Interior angle 900 ÷ 7 = 128.5°
• Question 2 Find the sum of the interior angles of this decagon.

Find the value of one of the interior angles.

 1440° 1260° 1800° 144° 126° 180° The sum of the interior angles is One interior angle is
EDDIE SAYS
10 sides - 2 = 8 triangles. 8 x 180 = 1440° 1440 ÷ 10 = 144°
• Question 3 This is an irregular octagon.

What is the size of the missing angle.

130
EDDIE SAYS
The only difference here is that this is an irregular polygon. To calculate the sum of the angles the principal is the same. 8 - 2 = 6 6 x 180 = 1080° As this is an irregular octagon, we have to add up the angles given and subtract from 1080. 150 + 160 + 140 + 100 + 170 + 120 + 110 = 950° 1080 - 950 = 130°
• Question 4 What is the size of an interior angle with 20 sides?

18
EDDIE SAYS
I hope you are not hiding behind your book. This time you want only the size of one of the interior angles. Find the total for the whole shape first. 18 x 180 = 3240° This time you want only the size of one of the interior angles. 3240 ÷ 180 = 18°
• Question 5 Here is an irregular heptagon.

What is the value of x?

59°
EDDIE SAYS
Here we go again, but we have got this. First find the sum of the angles. 7 - 2 = 5 5 x 180° = 900° Now simplify what we are given using our algebra skills. 2x + 2x + 10 + 2x + 2x -5 + 2x + 30 + 135° + 140° = 10x + 310 = 900° = 10x = 590 x = 59°
• Question 6 Can you help me fill in the blanks please.

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. 7 x 180 = 1260°
• Question 7 What is the value of x?

The value of x is 40°, 50° 60°
EDDIE SAYS
Whoah whats going on here. The meanies want us to use two sets of skills here. No worries lets go. A four sides shape the angles add up to 360° Simplify the algebra. 5x + 110 = 360 5x = 250 x = 50 and bingo.
• Question 8 Find the number of sides of a polygon with interior angles adding up to 1260&deg;

9
EDDIE SAYS
Just when you thought is was plain sailing....the curved ball. All we have to do here is work backwards. We know the formula to find the sum of interior angles is number of sides - 2 x 180 Instead take sum of the angles 1260 ÷ 180 = 7 7 + 2 = 9
• Question 9 Lets try another

Find the number of sides of a polygon with interior angles adding up to 8640°

50
EDDIE SAYS
Formulas are great aren't they. Once you know them you can always play around with them to help you. We know the formula to find the sum of interior angles is number of sides - 2 x 180 Instead take sum of the angles 8640 ÷ 180 = 48 48 + 2 = 50
• Question 10 Find the number of sides for the following.

 9 10 15 20 30 40 Interior angle 140° Interior angle 162° Interior angle 171°
EDDIE SAYS
Did you remember the rule for this one? Take the angle away from 180 and divide 360 by the answer. e.g 180 - 140 = 40 360 ÷ 40 = 9 Interior angles, sorted.
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