Area of Regular Polygons

Added: 2026-06-01

[00:00:01]So, this video is going to look at how we find the area of regular polygons.

[00:00:05]And regular polygons are polygons that are both equilateral, meaning they have equal sides, and equiangular, meaning they have equal angles. So, in a picture, you would see that all of these sides are congruent to each other. So, you'd see something like these dashes.

[00:00:19]And then you'd see that these angles are all congruent to each other, and that would be your heads-up that, "Hey, this is regular." So, we have here is we have a regular polygon with a circle circumscribed about it, meaning the polygon's vertices are all on the circle.

[00:00:33]We're going to use this to first define some vocabulary words, and then we're going to develop a formula. So, the first vocabulary word is the center. The center of the regular polygon is also the center of the circle that is circumscribing it, right? So, the center of the circle is the center of the polygon.

[00:00:50]The radius of the polygon is the distance from the center to a vertex.

[00:00:56]So, you could see I just drew two radii.

[00:00:59]The central angle, that's the um angle formed when you have two radii drawn to consecutive vertices. So, M and N, those are consecutive vertices. The central angle would be this over here, that angle there.

[00:01:16]And finally, this is the big one, the apothem.

[00:01:19]That is the perpendicular distance from the center to a side. So, that is this segment right here. That's the apothem, and it's the distance from the center to the side.

[00:01:32]And that's going to be important for our formula. So, the formula that you are going to need to know is that the area of a regular polygon is equal to 1/2 the apothem times the perimeter.

[00:01:44]So, yes, you're going to have to do probably quite a bit of trigonometry in order to figure out the missing distances that you need. Okay, so let's look at the first problem. Uh the first problem says, "Find the area of a regular hexagon with an apothem of nine units." So, a hexagon. So, a hexagon, we know has six sides. So, let me draw that. So, 1 2 3 4 5 6. It's regular, so all of these sides should technically be equal. I know they don't look it, but I'm marking them equal.

[00:02:18]And now let's draw our center.

[00:02:22]And let's draw two radii.

[00:02:25]The apothem is nine units. So, that tells you that this here, that segment there, is going to be nine units.

[00:02:33]All right. So, really what you have here is you have a triangle. And I'm going to draw this triangle off to the side. I'm going to draw this one right here. All right.

[00:02:42]So, you have your apothem of nine units.

[00:02:48]You have some angle up here.

[00:02:50]And then you have your radii. And this is 1/2 of the side length, right? Cuz the full side would be two triangles.

[00:02:57]This is half of it.

[00:02:58]So, the first thing is we have to figure out what that angle is up top there, okay?

[00:03:03]So, here's what you need to know.

[00:03:05]In a polygon like this, it would be 360° would be the entire kind of central angle there, right?

[00:03:14]We have a hexagon. And a hexagon, if I was to continue to draw these triangles, would have a total of 1 2 3 4 5 6 of them. Which means we would take our 360° divide them among six, and that would tell us that each of those big central angles there would be 60°. However, the triangle that I have drawn here would be half of that. Because the entire angle, that angle from here to here, would be 60. I only have half of that, which is 30.

[00:03:52]We also know, by definition that the apothem is perpendicular, so we know that's 90, which then means this is 60.

[00:04:00]So, special right triangle, right? The side opposite the 60 is 9, and we are interested in the perimeter. That's what we're missing.

[00:04:11]In order to get the perimeter, we need to know the side length. So, we need to figure out this bottom side here. So, let's use our special triangle relationship to solve that. So, opposite the 60 is x root 3, opposite the 30 is x, opposite the 90 is 2x.

[00:04:26]So, 9 = x root 3 divide by root 3, divide by root 3. So, we have 9 over root 3, which we could rationalize, right? So, 9 over root 3 * root 3 over root 3 9 root 3 over 3 simplify that, and you're going to get 3 root 3.

[00:04:48]So, that tells you then that the bottom side of this triangle is 3 root 3.

[00:04:56]However, that is not the full side length, right? Because right now, we just found this side down here, right?

[00:05:05]We found this here, which matches up with this here. So, we actually need to double that to find the full side length of the hexagon. So, our full side length is 6 root 3, because it's 2 * that. So, in order to find the perimeter, we would have 1 2 3 4 5 6 of those sides. So, our full perimeter is 6 * 6 root 3, also known as 36 root 3. That's our perimeter.

[00:05:37]Our apothem is 9.

[00:05:39]So, now we can plug it into the formula.

[00:05:41]The area is equal to 1/2 of 9 * 36 root 3.

[00:05:49]From there, we could simplify a little bit, right? So, we could cross out the two, make the 36 an 18, and our exact answer would be 9 * 18 root 3. So, 9 * 18 is going to give us 162 * the square root of 3.

[00:06:08]And that would be your final answer. 163 162 root 3 units squared.

[00:06:16]So, this is one where you're given the apothem, you're given the fact that it's a regular figure, and you have to use right triangle trig in order to solve for the missing side, which then allows you to find the perimeter.

[00:06:28]For this next question, uh we're finding the area of a regular polygon with nine sides and radius of 10. You can see I did my best to draw. Once again, this isn't our class, so you don't have to be perfect. You just have to give a somewhat reasonable representation.

[00:06:41]So, here is the center of our polygon.

[00:06:46]I could draw a triangle.

[00:06:48]Draw my radii.

[00:06:49]We know that the radius is 10, so this is 10, this is 10.

[00:06:54]Drop down the apothem.

[00:06:57]And now from here, let's figure this out, right? So, just like the last problem, we need to figure out first of all what the central angle is. So, remember, the entire way around would be 360.

[00:07:11]And we're going to divide that by the number of sides. So, in our case, we have nine sides, so 360 / 9 would give us 40.

[00:07:20]40° would be the entire angle. That would be this here.

[00:07:24]We don't want that, though. We only want half the angle. We want this here, cuz then we have a right triangle, which is half of 40, which is 20.

[00:07:32]So, now we have a 20° angle there, a 90° angle here, and then we could figure out what this angle is, right? So, that's going to be 90 + 20 is 110, and then 180 - 110 is 70.

[00:07:45]So now we could redraw that triangle and we could do some trigonometry.

[00:07:51]So here's this.

[00:07:53]This is 10. This is 20 degrees. This is 70 degrees.

[00:07:58]Um and then from there let's do some work. So first things first, we're going to have to figure out what the apothem is and we're going to have to figure out this which is half of the side, right? So let's start off with the apothem.

[00:08:12]So the apothem here we're going to use sine, cosine, tangent.

[00:08:19]Relative to the 20 degree angle, the apothem, which I'm just going to write down as A, is the adjacent. So we could say the cosine of 20 degrees is equal to A, the apothem or the adjacent in this case, over hypotenuse which is 10.

[00:08:37]Multiply both sides by 10, we get 10 cosine 20 degrees which as a decimal is approximately 9.

[00:08:47]397.

[00:08:49]Now we could do similar to figure out what half of the side length is. That's this here, right? Which is this.

[00:08:56]For that, we're going to have to use sine because that is the opposite. So we could say sine 20 is equal to the opposite over hypotenuse.

[00:09:14]So multiply both sides by 10, 10 sine 20 and then from there plug that in into the calculator and you're going to get 3.42.

[00:09:27]So remember that is half of the side of the nine-sided polygon. So to get the entire side, we're going to have to figure out 2 * 3.42 which is going to give us 6.84.

[00:09:47]Right? So, then that is going to be our entire side length.

[00:09:52]So, then to figure out what the perimeter is there are nine of those sides.

[00:09:58]So, 9 * 6.84 [snorts] is 61.56.

[00:10:05]We know the apothem is 9.3 97.

[00:10:10]We know perimeter, so area is equal to 1/2 apothem * perimeter.

[00:10:18]1/2 9.397 * 61.56 and plug it into the calculator and you're going to get a final area of approximately 289 square units. As you can see, these problems do get a little bit involved. However, just keep in mind, draw your polygon first.

[00:10:42]Figure out what the central angle is.

[00:10:43]Don't forget to divide it by two, right?

[00:10:46]In order to figure out what half of the angle is, you could draw your right triangle.

[00:10:50]Then from there, I'd suggest you draw the triangle on its own.

[00:10:55]And then solve for what you need, whether you need uh the side length, whether you need the apothem length and then just go from there. But, you're going to have to use SOH CAH TOA in order to figure out what sides you're missing.