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Polygon Puzzle: How Many Degrees Are in a Polygon?

How many degrees are in the interior angles of a pentagon? A hexagon? An octagon? Or any polygon? Keep on reading The Math Dude to learn how to solve this polygon puzzle!

By
Jason Marshall, PhD,
February 3, 2017
Episode #227

Page 1 of 2

Soccer BallWe recently talked about why the three interior angles of a triangle must always add up to 180º. And at various points in the past, we've noted that the quartet of 90º “right" angles in a square must mean that the interior angles of a square add up to 360º. But we've never talked about what happens when we toss more sides into the mix.

In other words, we've never talked about how to figure out the total number of degrees in a pentagon. Or a hexagon...or an octagon. Or any other polygon, for that matter! And just as importantly, we’ve never dealt with whether or not there’s some clever way to figure all of this out without resorting to making measurements with a protractor.

Until now, that is - because these are exactly the questions we’ll be talking about today as we dive into a delectably delicious polygon puzzler..

Review: Interior Angles of Polygons

Our big goal for today is to figure out exactly how the interior angles of polygons change as the number of sides in the shape increases.

As a super quick review, a polygon is any shape made up of three or more connecting sides that you can draw on a flat sheet of paper. For a more thorough look at the definition of a polygon, check out the episode on that topic.

The angles formed in the interior of a polygon where pairs of sides intersect are called "interior angles." As noted earlier, we've talked about using a clever trick to prove that a triangle's interior angles always add up to 180º. And we’ve seen that the four right interior angles of a square (or any rectangle) must add up to 360º.

Which might lead you to wonder...

How Many Degrees Are In a Pentagon?

What happens when the number of sides increases beyond four? In other words, what’s the sum of the interior angles of a pentagon, a hexagon, an octagon, or any other polygon?

Interior angles get larger as the number of sides increases.

Let’s start by taking a look at the 5-sided regular polygon (meaning, its sides and angles are all equally sized), better known as a pentagon. If you sketch a pentagon, you’ll immediately see that its interior angles are all greater than 90º. So the first thing we can conclude is that the interior angles of a polygon get larger as the number of sides increases. But by how much?

At this point, I encourage you to stop for a minute and see if you can figure out how you might go about answering this question. If you’re having trouble getting started, think about the fact that you can draw a diagonal line across a 360º square to break it up into a pair of 180º triangles.

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