Lesson Video: Exterior Angles of a Polygon Mathematics • 8th Grade

In this video, we will learn how to identify exterior angles of polygons, find their sum, and use them to solve problems.

11:37

Video Transcript

In this video, we’ll learn how to identify exterior angles of polygons, find their sum, and use these to solve problems.

We begin by defining the exterior angle of a polygon. It’s an angle at a vertex of the polygon outside the polygon, which is formed by one side and the extension of an adjacent side. We see from our diagram the exterior angle marked. Since these angles are made by extending a side and angles on a straight line add up to 180 degrees, we see that the corresponding interior and exterior angles are supplementary. That is, they add to 180 degrees. So, one way that we can calculate a single exterior angle in a polygon is to subtract the measure of the interior angle from 180 degrees. That isn’t always appropriate, though. So, let’s look at a way in which we can find the sum of the exterior angles in any polygon.

Olivia is experimenting with the exterior angles of a triangle. She colors the angles, cuts them out, and sticks them together as seen in the figure. Will angle 𝑧 fit in the space to complete the circle?

One way we can answer this question is to simply imagine the triangle getting smaller and smaller and smaller. We notice that as the triangle shrinks to nothing, these lines appear to circle around a point. We know that angles around a point sum to 360 degrees, so we can infer that angles 𝑥, 𝑦, and 𝑧 fit perfectly to complete the circle. And so, we could add in angles 𝑧 on our original diagram as shown.

But is there a better way to prove this? Well, yes, there is. We know the angles on a straight line are supplementary. They add to 180 degrees. We also know the same about interior angles in a triangle; they add up to 180 degrees. So, we’re going to fill in the missing three angles in our first triangle. This first one must be 180 minus 𝑥 degrees. This must be 180 minus 𝑦. And over here, we have 180 minus 𝑧 degrees. We know that the sum of these three interior angles is 180 degrees, so we can say 180 minus 𝑥 plus 180 minus 𝑦 plus 180 minus 𝑧 equals 180. This simplifies to 540 minus 𝑥 minus 𝑦 minus 𝑧 equals 180.

We’re going to subtract 180 from both sides so that 360 minus 𝑥 minus 𝑦 minus 𝑧 equals zero. And then we’re going to add 𝑥, 𝑦, and 𝑧 to both sides so that 360 equals 𝑥 plus 𝑦 plus 𝑧. We’ve actually proven that the exterior angles 𝑥, 𝑦, and 𝑧 do indeed add to 360 degrees. And so, angle 𝑧 will fit in the space and form a perfect circle. And so, the answer is yes. Note that we can generalize this and say that the exterior angles of any polygon sum to 360 degrees no matter the number of sides it has.

So, let’s look at how we can use this formula to find an exterior angle of a regular polygon.

What is the measure of the exterior angle of an equilateral triangle?

Let’s begin by recalling what we know about the exterior angles of a polygon. We know that the exterior angles of any polygon sum to 360 degrees no matter the number of sides it has. But we also know that all angles and sides in an equilateral triangle are equal.

Essentially, it’s a regular polygon. We can therefore assume that the exterior angles of our equilateral triangle must all be equal in size. There are three of them. So, to find the measure of one exterior angle of an equilateral triangle, we divide the angles’ sum, that’s 360 degrees, by three. 360 divided by three is 120. So, we can say that the measure of the exterior angle of an equilateral triangle is 120 degrees.

Now, we can generalize this process for any regular polygon. We can say that the measure of a single exterior angle of a regular 𝑛-sided polygon, that’s a polygon with 𝑛 sides, is 360 divided by 𝑛 degrees.

We’re now going to look at how we can find the number of sides of a polygon given the measure of a single exterior angle.

The measure of an exterior angle in a regular polygon is 40 degrees. How many sides does the polygon have?

Here, we have some information about the size of one exterior angle in a regular polygon. Remember, a regular polygon is one whose sides and angles are all equal. So, we recall that the measure of a single exterior angle in a regular 𝑛-sided polygon is 360 divided by 𝑛 degrees. And that’s because the total angle sum of the exterior angles is 360. Now, we don’t know how many sides our polygon has, so 360 divided by 𝑛 here gives us the measure of a single exterior angle, gives us 40 degrees. So, we can form an equation and say that 360 divided by 𝑛 equals 40. Let’s look to solve this equation.

We’ll begin by multiplying both sides by 𝑛. 360 divided by 𝑛 times 𝑛 is just 360, and 40 times 𝑛 is 40𝑛. So, our equation is 360 equals 40𝑛. We’re looking to solve for 𝑛. Remember, we’re trying to find the number of sides this polygon has. So next, we divide through by 40. So, 360 divided by 40 is equal to 𝑛. Well, 360 divided by 40 is nine. And so, this regular polygon must have nine sides.

We’ll now look at how we can use the fact that exterior angles of a polygon add to 360 degrees to solve problems involving irregular polygons.

The diagram shows an irregular quadrilateral. Calculate the value of 𝑥.

If we look carefully, we see that 𝑥 is the measure of an exterior angle. That’s an angle on the outside of a polygon formed by one side and the extension of an adjacent side as shown. In fact, we’re given two more exterior angles and then the measure of a third angle. But this is the interior angle; it’s inside the shape. And so, there are two ways we can answer this question. Let’s begin by recalling what we know about the exterior angles in a polygon. We know that the exterior angles in a polygon sum to 360 degrees. We also know that since the interior and exterior angles sit on single straight lines, a corresponding interior and exterior angle sums to 180 degrees.

And so, let’s begin by working out the exterior angle. Let’s call that 𝑦. We know that the sum of this and its corresponding interior angle is 180 degrees. So we can say that 𝑦 plus 110 is equal to 180. We’re going to subtract 110 from both sides of this equation. And when we do, we find that 𝑦 is equal to 70. And we now have three out of four of our exterior angles in our irregular quadrilateral. We know that the sum of all four must be 360 degrees. So, we can form a second equation, this time 70 plus 43 plus 92 plus 𝑥, that’s all our exterior angles, equals 360. 70 plus 43 plus 92 is 205. And so, we solve for 𝑥 by subtracting 205 from both sides. 360 minus 205 is 155. 𝑥 is equal to 155 or 155 degrees.

Now, in fact, there was a second way we could’ve answered this, but it’s a slightly more long-winded method. In this method, we begin by subtracting 43 and 92 individually from 180. In doing so, we calculate two more of the interior angles. Then, since the interior angles in a quadrilateral add to 360 degrees, we subtract the sum of 110, 137, and 88 from 360 to get 25. Then, 𝑥 is found by subtracting 25 from 180. Once again, that gives us 155 or 155 degrees.

Next, we’ll look at how we use the sum of the exterior angles of a polygon to find an unknown.

The diagram shows an irregular pentagon. Calculate the value of 𝑥.

Note here that this diagram is not drawn to scale. In our pentagon, we’ve been given one, two, three, four, five exterior angles. Remember, the exterior angles are angles at the vertex but on the outside of the shape, made between one side and an extension of the adjacent side. So, let’s recall what we actually know about exterior angles of polygons. No matter its number of sides, the exterior angles in a polygon all add to 360 degrees. So, let’s use this fact to form an equation.

We’ll add together the given angles from our diagram. That’s three 𝑥 plus three 𝑥 plus two 𝑥 plus two 𝑥 plus two 𝑥. And we know that adds to make 360 degrees. We simplify by collecting like terms, and our equation becomes 12𝑥 equals 360. Finally, we solve for 𝑥 by doing the inverse operation. At the moment, it’s being multiplied by 12. So, we’re going to divide both sides of our equation by 12. 12𝑥 divided by 12 is 𝑥. So, we need to work out 360 divided by 12. Well, 36 divided by 12 is three, so 360 divided by 12 is 30. And this means 𝑥 is equal to 30.

Now, where possible, we should try and check our solutions. Here, we can check by substituting our value for 𝑥 back into the original diagram. This angle will be three times 30, which is 90 degrees. Similarly, we have 90 degrees here. This angle will be two times 30 since it’s two 𝑥. That’s 60 degrees. And the remaining two angles are also two 𝑥, so they’re 60 degrees. The sum of these five values should be 360 degrees. So, let’s check. That’s 90 plus 90 plus 60 plus 60 plus 60, which is indeed equal to 360 as required. So, 𝑥 is equal to 30.

In this video, we learned that the exterior angle of a polygon is the angle at the vertex of that polygon on the outside of the shape. We learned that the sum of the exterior angles of a polygon is 360 degrees. And if we have a regular polygon, we can find the measure of a single exterior angle if it’s got 𝑛 sides by using the formula 360 divided by 𝑛. Finally, we saw that each pair of interior and exterior angles in a polygon are supplementary; they add to 180 degrees.

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