 In this video we're going to be talking about the exterior angles in a polygon. In addition to having interior angles, every polygon has a set of exterior angles at each vertex that are formed by extending a side and for a five-sided figure we would have five interior angles and five exterior angles. Before we get to the pros, let's just take a look at a regular triangle. Remember regular means that all sides are equal and all angles are equal. So a regular triangle, if the interior angles equal 180 degrees, we know each interior angle would have to be 60 degrees. We can form the exterior angles by extending the sides in the same direction all the way around that triangle. Each interior and exterior angle form a straight line and therefore form linear pairs and are going to be supplementary. So we can find the measure of each exterior angle by subtracting the interior angle from 180 degrees. Therefore for a regular triangle, each exterior angle would equal 120 degrees. If we wanted to add up the exterior angle to find the sum of the exterior angles, that would just be 120 times the three exterior angles because there's three sides. That would give us a sum of exterior angles of 360 degrees for any regular triangle. If we go ahead now and try this with the problems that we have, we can use the same thinking for our pentagon for our first problem. We found before that an interior angle of a pentagon is 108 degrees. That's because the sum of the pentagon, 540 degrees, is evenly distributed between the five interior angles. If we know that each of these interior angles is 108 degrees, we can do the same thing we did with the triangle and find the exterior angle by subtracting 108 from the 180 because these two angles will form a linear pair. If we know that each exterior angle of this pentagon is 72 degrees, to find the sum of the exterior angles, we're just going to do 72 times the five exterior angles. If you put that in your calculator, we're going to get, again, 360 degrees is the sum of the exterior angles. We notice that this measure 360 degrees for the sum of the exterior angles of a pentagon happens to be the same amount as we found for the triangle. In fact, no matter the number of sides of a polygon, the sum of its exterior angles will always equal 360 degrees. That's because as you go around the polygon, the more sides you have, the smaller that exterior angle gets, but they're always going to, if you put them together, complete a circle and we know a circle has 360 degrees. So any problem, no matter the number of sides of our polygon, if we're looking to find the total or the sum of the exterior angles, it's always going to be 360 degrees. We don't have to draw a 15-gon. We don't have to calculate the interior sides to find the exterior sides. In fact, our next problem here, asking us for the n-gon, that's asking us for the formula to put in our formula box for any polygon regardless the number of sides. And we know that no matter how many sides, exterior angles of any polygon will always equal 360 degrees. So that's the official formula for the sum of exterior angles. While the sum of the interior angles is going to be dependent on the number of sides, the exterior angles will always equal 360 degrees. The next set of problems asks us to now find one exterior angle of a regular polygon. We already did that with our pentagon in the example above and found out that that exterior angle, each exterior angle was 72 degrees and we found that by realizing that the sum of the interior angles, 360 was just equally distributed between those five exterior angles to get 72 degrees. And you can probably get the idea then, we will do for a 15-gon, we know the total sum of the interior angles is going to be distributed equally over 15 sides. And when we put that in our calculator, that comes out to 24 degrees. That's our formula for any sided polygon. We're going to take the sum of the exterior angles and divide it by its number of sides. So we'll go ahead and fill in the last box of our formula cube here. We know the sum of the exterior angles will always be 360, no matter the number of sides. And if we're looking to find just one, we're going to take that 360 and divide it by n. So as you're going through your problems and your homework and your classwork, it's important to realize what they're asking you, if they're talking about interior angles versus exterior angles and whether they're asking you for the sum or the total of the angles versus just one individual angle. And if you understand the way the formulas relate to each other, this will come in handy.