Interior and Exterior Angles of a Polygon

Interior and Exterior Angles of a Polygon


Welcome to a lesson on angles of convex polygons. The goal the video is to determine the sum of the interior and exterior angles of a polygon. Every convex polygon has interior and
exterior angles. The interior angles are the angles inside the polygon formed by the sides. So for this pentagon
there are five interior angles. Angle one, angle two, angle three, angle four and angle five. The exterior angles form a linear pair with the interior angles. So if we look at angle two for a moment, we can either extend this side here, to form the exterior angle, or we can extend this side here to form the exterior angle. It doesn’t matter which side we extend to form our linear pair. Let’s go ahead and extend this side here. And now I’ll go ahead and work my way around the pentagon extending each side. So we’ll extend the side here, this side here, this side here, and the side here. Notice we have a five sided polygon, and I extended five sides to form the
exterior angles. So looking at angle one, angle six forms a linear pair with angle one, and therefore is an exterior angle. Here we’d have angle seven, angle eight, angle nine and angle ten. Each of the blue angles form a linear pair with the red interior angles. And before we talk about the sum of the interior and exterior angles, let’s review the sum of the interior angles of a triangle. Remember the sum of angle one, angle two, and angle three is always equal to one hundred eighty degrees. And just to demonstrate this, let’s go ahead and take a look at a wolfram demo. Here we have a triangle. If we determine the midpoints of side AB and side AC, they would be here. If we connect the mid-segment, we would have this. And then we’ll construct the perpendiculars from these midpoints to the base. And now we’ll make folds across these segments. One, two, three folds. And now we see very clearly that angles AB and C would form a straight angle, or would have a sum of one hundred eighty degrees. So let’s use that information to determine the sum of the interior and exterior angles of some convex polygons. So for this
first triangle we know the sum of the interior angles, angle one, two and three, would be one hundred eighty degrees. Now let’s go ahead and label the exterior angles. So we’ll extend each side starting here. And so the exterior angles would be angle four, five and six. Well the sum of the interior and exterior angles would be one hundred eighty degrees times three, since angles one and four, three and five and two and six form linear pairs, we know their sums
would be one hundred eighty degrees. So this would give us five hundred forty degrees and if we subttract out the sum of the interior angles, or one hundred eighty degrees, we’d have the sum of the exterior angles. That’s going to give us three hundred sixty degrees. Now for the interior angles of a
quadrilateral we’re going to break up the interior into triangles since we know
the sum of the interior of a triangle would be one hundred eighty degrees. Using that logic we can see that there would be two triangles in the interior of a square, and therefore, the sum of the interior
angles would be one hundred eighty degrees times two, or three hundred sixty degrees. And then when it comes to the exterior angles, we’ll go ahead and extend each side. So if we the label the interior angles and the exterior angles, the interior and exterior angles make up four linear pairs. So the total
sum would be one hundred eighty degrees times four, so that’d be seven hundred twenty degrees. And if we subtract out the sum up the interior angles, which is three hundred sixty degrees, we would have three hundred sixty degrees as the sum of the exterior angles. So notice how the sum of the interior angles is changing, but the sum of the exterior angles is
staying the same. Let’s go ahead and try at least one more. For the pentagon, we can divide the interior into three triangles. Therefore, the sum of the interior angles would be one hundred eighty degrees times three, or five hundred forty degrees. And if and there are five interior angles there would be five exterior angles, so we’d have five any repairs. So the sum of the interior and exterior angles would be one hundred eighty degrees times five, it’s going to be nine hundred. Subtracting out the sum of the interior angles again, leaves us with three hundred sixty degrees. So once again, notice that the sum of the interior angles is increasing while the sum of the exterior angles stays the same. Let’s go back and see if we can notice
a pattern. Notice that the polygon has three sides, we can think of this as one hundred eighty times one. For four sides we have one hundred eighty degrees times two, and for five sides we have one hundred eighty times three. So you can probably see the pattern, which leads us to the following theorem. The sum of the measures of the interior angles of a convex polygon with n sides is n minus two, times one hundred eighty degrees. So we take the number of sides, subtract two, multiply by one hundred eighty degrees and it gives us the sum of the interior angles. And the sum of the measures of the exterior angles of a convex polygon is always three hundred sixty degrees. Let’s take a look at an example using
these theorems. Here we want to determine the measure of each interior and exterior angle of this regular polygon. Remember if its regular, the lengths of all the sides and the
measure evolving interior angles are equal. So let’s first first count the number of sides we have. One, two, three, four, five, six, seven. So it’s a heptagon and n is equal to seven. So we’ll start be finding the sum of the interior angles. Using our formula we’re going to have n minus two, well there are seven sides here, so we’ll seven minus two, times one hundred eighty degrees. That’s going to give us five times one eighty, that’ll be nine hundred degrees. So the measure of each interior angle would be nine hundred degrees divided by seven, since we have a regular polygon. Here we have to perform long division. It’d be one hundred twenty-eight and four-sevenths degrees. Let’s go ahead and show where this came from. Nine hundred divided by seven, that’d be one, subtract, that’s two, bring down the zero. How many sevens in twenty? That would be two, subtract that’s six. Bring down the last digit. How many seven hundred sixty? That’d be eight, with the remainder of four. So we put four over seven for four-sevenths. Now for the sum of the exterior angles, we now know that it’ll always be three hundred sixty degrees. And since we have a regular polygon the measure of each exterior angle will also be the same. So we’d have three hundred sixty degrees divided by seven, that’ll be fifty-one and three-sevenths degrees. And again, let’s go ahead and show that as well. So we’d have three hundred sixty divided by seven. There’s five sevens in thirty-six, and there’s one seven in ten, so we have fifty-one and three-sevenths. Now remember, keep in mind this only works because we know it’s a regular polygon. If it’s not regular, each interior and
exterior angle may not be the same. I hope you found this helpful. Thank you for watching.


11 thoughts on “Interior and Exterior Angles of a Polygon

  1. Anyone else looking at the fact that this video has 35 000 views but less than 100 likes

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