Angles in Polygons

You have probably heard many times that Angles in a triangle add up to 180 degrees and that angles in a quadrilateral add up to 360 degrees. If you haven’t, this is your reminder that angles in a triangle add to 180 degrees and angles in a quadrilateral add to 360 degrees. However, have you ever wondered what angles in a five, six or even seven-sided shape sum to? What if we had a 24 sided shape? Okay, you have probably not. Regardless, in this article, we will be exploring angles in polygons. However, we must first outline what we mean by ‘polygon’.

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Last Updated: 01.12.2023
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The term ‘poly’ means many, so a polygon is just a shape with many sides. When we say ‘many’, we mean three or more. So essentially, a polygon can be any 2D shape that's not a circle. A polygon is a regular polygon if all sides and Angles are the same.

Interior Angles in Polygons

When we talk about what angles add up to a polygon, we refer to the sum of interior angles. We will use this term a lot from now on, so it is essential to know it.

Angles in Polygon, Polygon with Interior angles Labelled, Jordan Madge Angles in Polygons- Polygon with interior angles labelled, Jordan Madge- StudySmarter Originals

For a polygon, an interior angle is an angle inside the polygon (see the diagram above). The sum of interior angles is what all of the angles inside the polygon add up to. So, formally, we already know that the sum of interior angles in a triangle is $180 °$ and in a quadrilateral is $360 °$ .

Sum of Interior Angles Formula

Previously, we have just been expected to know that the interior angles in a triangle sum to $180 °$ and the interior angles in a quadrilateral sum to $360 °$ . We have just taken it as a fact and have never really questioned it. However, you may now be thinking, why is this the case? Or you may not... However, a convenient formula tells us the sum of interior angles for any polygon. It goes as follows...

For any given polygon with n sides,

$S u m o f I n t e r i o r A n g l e s = (n - 2) \times 180^{°}$

So, when we have a triangle, $n = 3$ and so the sum of interior angles is $(3 - 2) \times 180 = 180 °$ .

Similarly, when we have a quadrilateral, $n = 4$ and so the sum of interior angles is $(4 - 2) \times 180 = 360 °$

We already knew those two results. However, now we can apply this formula to shapes with more than four sides.

Calculate the sum of interior angles for a pentagon.

Solution:

A pentagon has five sides, so using the formula, the sum of interior angles is $(5 - 2) \times 180 = 540 °$

Calculate the sum of interior angles for a nonagon.

Solution:

A nonagon has nine sides, so using the formula, the sum of interior angles is $(9 - 2) \times 180 = 1260 °$

Calculate the sum of interior angles for the shape below.

Angles in Polygons, Example with 14 sided polygon, Jordan Madge Angles in Polygons- 14 sided polygon, Jordan Madge- StudySmarter Originals

Solution:

The above shape has 14 sides and so the sum of interior angles is $(14 - 2) \times 180 = 2160 °$

Calculate the sum of interior angles for a 24 sided shape.

Solution:

When id="2555885" role="math" $n = 24$ , the sum of interior angles is $(24 - 2) \times 180 = 3960 °$

Calculate the size of the angle x in the image below.

Angles in Polygons, Example with quadrilateral, Jordan Madge Angles in Polygons- quadrilateral example, Jordan Madge- StudySmarter Originals

Solution:

This shape has five sides, so the sum of interior angles is $(5 - 2) \times 180 = 540 °$

Each of the right angles in the shape is $90 °$ and so we can work out the missing angle by subtracting all of the given angles from 540. Thus, $x = 540 - 90 - 90 - 90 - 130 = 140 °$

Table of Common Interior Angles

The below table shows the sum of interior angles for the first eight polygons. However, you could confirm these results for yourself using the formula.

Shape	# sides	Sum of Interior Angles ( $°$ )
Triangle	3	180
Quadrilateral	4	360
Pentagon	5	540
Hexagon	6	720
Heptagon	7	900
Octagon	8	1080
Nonagon	9	1260
Decagon	10	1440

Calculating Each Interior Angle

Earlier, we defined regular polygons as polygons with equal sides and angles. We therefore may wish to calculate each interior angle of a regular polygon. We first calculate the sum of interior angles and divide this Number by the Number of sides.

Calculate each interior angle for a regular hexagon.

Solution:

Using table 1, we can see that the sum of interior angles for a hexagon is $720 °$ . Since this hexagon is regular, each of the angles is the same and thus we can work out each interior angle by dividing $720$ by $6$ . Therefore, each interior angle is $120 °$ .

Below is part of a tiling pattern consisting of three regular pentagons. Calculate the angle labelled x.

Angles in Polygon, Pentagon Example, Jordan Madge

Angles in Polygons- Pentagon Example, Jordan Madge- StudySmarter Originals

Solution:

The sum of interior angles for each regular hexagon is $720 °$ (using the table of common interior angles).

Thus, each interior angle in each hexagon is $120 °$ .

Angles in Polygon, Pentagon Example, Jordan Madge Angles in Polygons- Pentagon Example, Jordan Madge- StudySmarter Originals

Recall that angles around a point sum to 360 degrees. Therefore, x can be found by subtracting the other known angles from 360. Thus, $x = 360 - 108 - 108 = 144 °$

Exterior Angles in Polygons

There is also an exterior angle for each interior angle in a polygon. An exterior angle is formed between any side of the shape and the straight line extended outside of the shape. This may sound not very clear, but it is easier to see illustrated.

Angles in Polygon, Polygon with Interior and Exterior angles Labelled, Jordan Madge Angles in Polygons- Pentagon with interior and exterior angles labelled, Jordan Madge- StudySmarter Originals

In the diagram above, the interior angles are labelled orange, and the exterior angles are green. Since the exterior angle lies on the same straight line as the interior angle, the sum of the interior and exterior angles is $180 °$ . Therefore, an exterior angle can be calculated by subtracting the interior angle from $180 °$ .

In the image below, angles x and y are exterior angles. Calculate x and y.

Angles in Polygons, Example with exterior angles, Jordan Madge Angles in Polygons- Pentagon with interior and exterior angles, Jordan Madge- StudySmarter Originals

Solution:

For exterior angle x, the interior angle is $109 °$ . Thus, since angles on a straight line add up to $180 °$ , $x = 180 - 109 = 71 °$ . Angle $y$ is another exterior angle and since angles on a straight line add to 180, $y = 180 - 81 = 99 °$ .

Calculate each exterior angle of a regular heptagon.Solution: A heptagon has seven sides and thus the sum of interior angles is

900 °

Since this heptagon is regular, we can work out each interior angle by dividing 900 by 7 to obtain

128.6 °

. Therefore, we can calculate each exterior angle by subtracting this from 180. Thus, each exterior angle is

180 - 128.6 = 51.4 °

A heptagon is also sometimes referred to as a septagon.

Sum of Exterior Angles

The sum of exterior angles for any polygon is dead simple. It is $360 °$ . Unlike interior angles, we do not need to memorise any fancy formulae to work out the sum of exterior angles; we simply need to remember the sum of exterior angles for any polygon $360 °$ . Using this, we can begin to answer some more questions.

Each exterior angle of a regular polygon is 10. Work out the number of sides the polygon has.

Solution:

Since the sum of exterior angles is $360 °$ , and each exterior angle is $10 °$ , we can calculate the number of sides by $360 \div 10 = 36$ . Thus, this polygon has 36 sides.

Each interior angle of a regular polygon is 165. Work out the number of sides the polygon has.

Solution:

If each interior angle is $165$ , each exterior angle must be $180 - 165 = 15 °$ . Since the sum of the exterior angles is $360 °$ , there must be $360 \div 15 = 24$ sides.

Angles in polygons - Key takeaways

The interior angles in a polygon are the angles inside the polygon.
To calculate the sum of interior angles, subtract two from the number of sides and multiply the result by 180 degrees.
If the polygon is regular, each of the sides is the same.
An exterior angle is formed between any side of the shape and the straight line extended outside of the shape.
The sum of exterior angles of any polygon is 360 degrees, regardless of the number of sides.

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Frequently Asked Questions about Angles in Polygons

What do angles in a polygon add up to?

It is different for each polygon. The sum of interior angles in a regular polygon can be found by subtracting two from the number of sides and then multiplying this result by 180 degrees.

What is the sum of the exterior angles of a polygon?

The sum of exterior angles is 360 degrees for any polygon.

What is the formula for the sum of the interior angles of a polygon?

(n-2) x 180

What is the sum of the interior angles of a polygon?

The sum of interior angles in a regular polygon can be found by subtracting two from the number of sides and then multiplying this result by 180 degrees.

How to find the missing angle in a polygon?

First work out what the sum of angles should be, and then subtract the angles that you know to work out the missing one.

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