 Hi, this video is called area of a regular polygon formula. So by the time this video is done, you will have developed a formula for area of a regular polygon and done your first example using the formula. This first slide is dedicated to helping you see where the formula comes from. Instead of just giving it to you and asking you to trust me, you'll actually be able to see its development. We are going to use this regular pentagon. So by regular, we know all five sides are the same. We're going to use it to help us find our formula for area of a regular polygon. So even though I'm using a pentagon in my example, this formula will hold up for any regular polygon. So any sort of a shape that has all congruent sides and all congruent angles. Alright, where to begin? I have my pentagon. I'm going to look at the shape underneath it. I've got a parallelogram up on the slide. You all have learned that area of a parallelogram is base times height, where the base and the height are perpendicular to each other. They meet at 90-degree angles. We're going to use that now to help us with our formula. So let's think about this. If I look at my pentagon for a minute, I've got one triangle sitting inside of it. I got that triangle by drawing in radii. Let's go ahead and draw in all five radii that a pentagon will have. And you can see that a pentagon that has five sides has been fairly easily broken into five congruent triangles. So why don't you go ahead and draw in the radii and then just kind of lightly shade in your five triangles. What I'm going to do is kind of pretend like I cut this pentagon into five triangles and see if I can fit it into the parallelogram below. So here we go. This is what's going to happen. I'm going to make my first triangle, move my first triangle down right here. I have four more to go. My second triangle, I'm going to put it right next to it. My third, my fourth, and my fifth. So what I've done is I've taken the five triangles that were in my pentagon and rearranged them so they're sitting nicely in that parallelogram. I did that because I know the formula for area of a parallelogram is base times height, so I'm going to use that to help me, okay? So let's look at this. In my parallelogram, how much of the parallelogram was filled up? I have one, two, three, four, five triangles that were filled up and one, two, three, four, five that were not. So if I know that the parallelogram is base times height, I think my pentagon will be base times height, but only half of that. So one half base times height, or you could say base times height, divide by two. Mathematically, it's more proper to follow the formula one half base times height, but base times height divide by two is the same thing because multiplying by one half and dividing by two are the same. So you can see that we find the area of our pentagon by finding the area of a parallelogram and either multiplying by one half or divide by two. But now here's the thing. I don't want to have to do that every time where I take my shape and see how it fits into a parallelogram. So let's change our formula, and instead of using it in terms of the parallelogram using the base and height, could I rewrite it a little bit so it's in terms of my pentagon? So let's think about this. So what I'm trying to do is instead of using b and h, I want to use letters that come from my pentagon picture to help me. So looking at this, instead of calling the height h, what could I call it? Well remember, if this is the base of my parallelogram, the height will be a distance that meets it at a 90-degree angle that's perpendicular. So this could be considered the height. This could be considered the height. This could be considered the height. This could be considered the height, and so could this. So instead of calling it h, let's call it a for that apathome or apathome. Okay? Now what could I rename the base? Well the base of my parallelogram was this entire bottom side, which looks like it was the bases of all five of my triangles. So if I go back up in the picture, the bases of all five of my triangles are the distance around. What else could you call that? Well, the bases of the parallelogram, being the bottom five bases of my triangle is actually the perimeter of my pentagon. So instead of calling it the base of the parallelogram, let's call it the perimeter of my pentagon. So my formula, my official formula for area of a regular polygon is one-half perimeter times apathome, where it's the perimeter of the pentagon and the apathome or the apathome of my shape. So we could say one-half p times a, or p times a divided by two. Remember both formulas are fine, but the one-half p times a is more mathematically proper. I personally don't care which one you use. Ask your teacher if they have an opinion. This is Mrs. Milton. So if you don't have Mrs. Milton, ask your teacher if they care which formula you use. All right, this video isn't quite done yet. Now that we have our formula made, let's just quickly go through one example so we can use the formula. When this video is done, you will be done with the front side of your note sheet. We are now in the block that says number two. Find the perimeter of the regular polygon. Well, this is just giving us a little bit of practice. If it's a regular polygon, that means all five of these sides are the same. Since one of them is marked with a six, they all will be the six. So what's the perimeter? You could say six plus six plus six plus six plus six, or you can say six times five. You're going to get a perimeter of 30 units. All right, let's go to the next slide. Again, we're just practicing little pieces and we're going to put it all together really soon. Number three on the front side says find the central angle or CA in the regular polygon. Well, remember the central angle goes from the center of your pentagon, I'm sorry, out to the vertex. So since a pentagon has five sides, we'll have five radii. So that will make five central angles that are all the same. Well, like we mentioned before, all those central angles should add up to 360. And if they're all the same, they're going to divide by five, and that gives us 72 degrees. So I know all five of these central angles are 72 degrees. Okay, again, just one piece to the puzzle. We're practicing all the different parts. We'll put it together soon. The next one, number one, or number four, it says find the length of the apathem or apathem in the regular pentagon. Well, I'm going to just catch the picture up a little bit. All of these central angles were 72 degrees. And remember how all of these sides were six. We talked about what the apathem or the apathem does to the side of your pentagon. When this apathem dropped down, it split that into two, so we really had three and three here. So now I've got to kind of think about what could I do to find that apathem? Well, remember, in my pentagon, I've got five triangles going, and that apathem in that bottom triangle, it cut my triangle in half into two separate triangles, and it made a 90-degree angle. So sure enough, we have got a right triangle. So I'm taking this guy right here and making him bigger. We have a right triangle where this bottom side length was three. Here's the apathem that I'm trying to find. And what happened to this central angle that was 72? Well, when you drop an apathem down and it cut the segment, it bisected the segment into three and three, it also bisected the angle. So the 72 got cut in half to create 36. So we also have two angles that are both 36 degrees. So up here, I've got a 36, I've got a 90, and then I would also have... this would end up being a 54 because 36 plus 90 plus 54 is what adds up to 180. So now, hopefully, where everything's been pretty new, this is starting to look familiar, that we actually have Sokotoa. It's not a 30, 60, 90. It's not a 45, 45, 90. So I'm going to use Sokotoa to figure out what this apathem length is. So let's try to remember, I'm going to go... you could go from the perspective of the 36 or the 54. I'm going to go from the 54. It makes the apathem my opposite and three my adjacent. Well, opposite over adjacent is a tangent. So I've got the tangent of 54 equals the opposite, which is A, for apathem over the adjacent, which is three. I'm going to put the tangent of 54 over one. I'm going to cross-multiply. A times one is A, the tangent of 54 times three. Well, the A is already alone, so all I have left is to multiply. When I multiply those, I get it to round to 4.1. So my apathem is 4.1. All right, so now it's time to put it all together. You can see on this last slide on the front page, I've got a picture of my pentagon with the side length of six. I just found that apathem length in my last slide to be 4.1. So let's find the area of the regular pentagon. So it's time to put it all together. My formula that I learned is one-half perimeter times apathem. So I need to replace the P with the number and the A with the number. I just got the apathem to be 4.1. And if you can remember, we've already gotten the perimeter to be 6 times 5, which was 30. So now it's just time to simplify. Since it's a multiplication, one-half times 30 times 4.1 can multiply it in whatever order I want. I'm just going to go left to right. One-half times 30, that's the same thing as 30 divided by 2 is 15. 15 times 4.1 gets me 61.5 units squared. So you have finally, in your longest video yet, just completed a problem to find the area of the regular pentagon. There's some things we're going to have to talk about with rounding. We'll deal with that in some later problems to really concentrate on that part. But for now, you've learned how to use one-half perimeter times apathem. You've just finished your first problem. Now it's time to do a lot more so you can get good at this.