 In this video, Interior Angles of a Polygon, we are going to use some information we know about common polygons to derive a formula to find the sum of angle measures for any polygon regardless of the number of sides. So we'll start by filling in some information that we know. We know a triangle has three sides and we know of course that any triangle is always going to have the sum of its angle measures equal 180 degrees. If we move on to a quadrilateral, it might not be as well known the sum of the angle measures of a quadrilateral but we can by creating triangles inside this polygon going from vertices to vertices. We've created two triangles and we know that each of those triangles, the sum of the angle measures will be 180 degrees each. So for any four-sided figure, the sum of its angle measures is 180 times 2 which is 360 degrees. We can go ahead and move on a pentagon and do the same thing to find the sum of its angle measures. Take any vertices and go to another one. We've created one triangle, two, three triangles. Each of these triangles is 180 degrees. So for any five-sided figure, any pentagon, we're going to have 180 times 3 which if you have your calculator out is 540 degrees. We'll do this one more time if you want to draw a hexagon. A hexagon is any six-sided figure and again I'm going to create triangles and it doesn't matter how you do it as long as you create three-sided figures in there. And we know that each triangle is 180 degrees and in a hexagon we have four triangles. Four times 180 degrees is 720. We could keep on going but hopefully we can notice a pattern here between the number of sides and the sum of the interior angles in the polygon. We saw that when we had three sides we had one triangle and so it was 180 degrees. With four sides we created two triangles and two times 180 is 360 degrees. Five sides gave us three triangles which was 540 degrees, 180 times 3 and lastly we did a hexagon which created four triangles. Four times 180 was 720. We can notice that each time we go up a number of sides we go up one triangle and the difference here, 6 minus 4, 5 minus 3, 4 minus 2 and 3 minus 1, we're just multiplying by 2 less than the number of sides and that is going to give us our formula, the number of sides in any polygon. We know no matter how many sides our polygon has, if we subtract 2 from that number, that's going to be how many triangles we can create in that. That will help us out as we go forward because if we get a 23-sided figure, for example, we don't want to draw and separate it into triangles, we can just plug it into this formula to find out the sum of the interior angles. You don't have to write this down but this is just to recap that 3-sided figure then n equals 3, we would just plug that into our formula and minus 2 times 8, that gives us the sum of the interior angles if you added up the interior angles of any 3-sided figure. Again for any 4-sided figure, any quadrilateral, n equals 4, you're just going to plug in the number of sides multiply it times 180 degrees to get the sum of the interior angles. So you can go ahead and fill in this box, we've done our first formula, our 1 out of 4 formulas that we're going to learn in our videos today, and you might want to jot down some of the names of the common polygons that we'll be working with this chapter. You certainly don't have to draw the pictures and we know a lot of these, we know a 3-sided figure as a triangle, 4-sided figure quadrilateral, 5-sided is pentagon, 6-sided is hexagon the couple that come up that maybe aren't as familiar is the 7-sided figure, which is a heptagon you might want to jot that down, 8 is octagon 9 is nonagon, and a 10-sided figure is a decagon. If you don't know those go ahead and make yourself a note because those will come up These last two for an 11-sided figure and a 12-sided figure, you don't need to know those, but just extra information, 12-sided figure dodecagon. Usually anytime we go above a 10-sided figure, we're just going to take the number of sides and add the gun. So a 12 gun is the same thing as a dodecagon. If we're talking about a 54-sided figure, we would call it a 54 gun.