Sum of Angles in a Polygon

This chapter explores sum of angles in a polygon. The objectives of this chapter are; to be able to understand and use the formula for the sum of interior angles of a polygon with n sides, and be able to calculate the number of sides of a polygon using information about its angles. No prior knowledge is required for this chapter.

Interior angles of pentagon

Below is the pentagon. Suppose we wanted to find the sum of the interior angles.
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To find the sum of angles we would choose a central point and label it C and then divide the pentagon into 5 triangles as shown below.
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Above you can see that we have formed triangles when we divided the pentagon into 5 triangles. We know that the angles in each of the triangles add up to 180°
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Then sum of all the angles in all the triangles is;
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We can see that the total is made up of the angles around the point and the angles are labelled with blue curve that is;
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The angles at C make a complete turn which means they add up to 360°
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The blue angles are known as the interior angles S;
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…that is;
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Interior angles of an n-sided polygon

This section explores how to find the sum of the interior angles of an n-sided polygon. We can use some of the skills we gained in the previous example for this practice.
The diagram below shows part of an n-sided polygon;
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As we did above we divide the polygon into n triangles as shown below.
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It is very obvious that the sum of all the angles in all of the triangles will be;
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…since a triangle has a total of 180° when all the angles are added up. The angles marked with a red circle below make a complete turn and therefore add up to 360°.
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The blue angles shown below are known as the interior angles.
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From above we can form a generalised formula.
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The following examples should help revise sum of interior angles.

Example 4

Find the sum of interior angles of a polygon with 20 sides.
We know that the sum of the interior angles is given by;
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If we substitute in n with 20, we get;
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Example 2

A regular polygon has 12 sides. Find the size of each interior angle.
First we find the sum of the interior angles. This is given by;
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We substitute n with 20 which gives;
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We know that the polygon is regular this means that each angle is the same size. We also know that there are 12 interior angles.
Each interior angles is 150° because;
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Finding the number of sides

The diagram below shows part of a regular polygon. Suppose we wanted to add the number of sides the polygon has;
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We have an exterior angle shown as 20°. Since it is a circle all interior angles must add up to 360°. We know that all the exterior angles are equal so if the polygon has n sides then;
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We divide both sides by 20 to get;
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The polygon must have 18 sides.
We can work this out since the polygon is regular. We can also work out the number of sides if we know the sum of the interior angles.

For example

The sum of the interior angles of polygon is 1980°. How many sides does it have?
We can use the “sum of the interior angles” formula. The formula will still work for any polygon it doesn’t need to be a regular polygon. If the polygon has n sides then;
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We divide both sides by 90° which gives;
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Simplify;
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The polygon must have 13 sides.