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’.

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 Polygons- Polygon with interior angles labelled, Jordan Madge- StudySmarter Originals

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,

$SumofInteriorAngles=(n-2)\times {180}^{°}$

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.

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.

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.

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

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 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 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 $180°$.

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.