Area of regular polygon given side length
About this video
The video finds the area of a regular pentagon with side 13 using Area = 1/2 * apothem * perimeter. It splits the pentagon into 10 right triangles, uses a 36-degree central angle and tangent to get the apothem 6.5/tan(36), and gets an area of about 290.76 square units.
You'll be able to
- explain what an apothem is and why Area = 1/2 * apothem * perimeter works for regular polygons.
- find the central angle of the right triangles inside a regular polygon.
- use tangent to find an apothem from a side length and calculate the polygon's area.
Key words
- regular polygon
- A polygon whose sides are all the same length and whose angles are all equal.
- apothem
- A segment from the centre of a regular polygon that meets a side at a right angle, cutting it in half.
- perimeter
- The total distance around the outside of a shape.
- tangent
- In a right triangle, the ratio of the side opposite an angle to the side adjacent to it.
Transcript
- [Instructor] So what I have over here is a pentagon. It's a regular pentagon, so all five sides have the same length, and we know that the length of each of those five sides is 13. The goal here is to try to figure out the area of this pentagon. So pause the video and have a go at it before we do this together.
Okay, now let's do this together. So just to remind ourselves, if we want take the area of a regular polygon like this, it's going to be equal to 1/2 times the apothem, I've sometimes seen this, let me make sure I'm on the right layer, times the apothem length, which I'll talk about in a second, times the perimeter.
Now, first of all, what is an apothem? I didn't say an opossum, an apothem. Well, the apothem is any line that goes from the center of the polygon and intersects one of the sides at a right angle. So, for example, this right over here is an apothem. This right over here is also an apothem.
Let me draw it a little bit clearer like that. And it bisects the side that it intersects. I'm not proving it here, but it does actually do that. Now, before I even try to figure out what the apothem length is and the perimeter, why does this make sense? Well, it makes sense if you think about it, this has a lot of similarity between, or a lot of similarity with the area of a triangle, and that's not a coincidence because you can divide any regular polygon into a series of triangles.
In this case, you could divide it into five triangles. And what would be the area of each of these triangles? Well, it would be 1/2 times the height, which is the apothem length, times the base. And then if you wanted the areas of all of them, well, you would have five of these bases, and so another way to think about it is it would all boil down to this formula right over here.
You can try that if you like, but let's get into it and actually try to figure out the apothem length because they don't give it to us here, they just give us the length of one side. And to figure that out, we're gonna have to figure out a little bit of trigonometry.
So the first thing to realize is if we were to go all the way around, that would be 360 degrees. But if we wanted to just figure out, whoops, if we just wanted to figure out this angle here, what would that be? Well, that is 1/10th of the way around. One way to think about it, let's divide all of these or draw some apothems on all of these triangles here.
And you could see that there's one, two, three, four, five, six, seven, eight, nine, 10 equal sections that we are divided into, so the angle is going to be divided into 10. So that 360 degrees, 1/10th of that is going to be a 36 degree angle. Now, why is that useful?
Well, we know this angle, do we know the length of this side here? Well, this side here is gonna be exactly half the length of that side, so that's going to be 13 over 2 or let's just write that as 6.5. And then we could use that to figure out the apothem length.
Let me write it over here. The apothem, let me just call that, well, maybe I'll call that x just to not have to keep writing that. So we're going to, x is going to be equal to what? Or actually maybe another way to think about it is if I'm dealing with the opposite and the adjacent side to the angle, that sounds like tangent could be useful.
Tangent is opposite over adjacent. So I could say that the tangent of 36 degrees, tangent of 36 degrees is going to be equal to 6.5, the length of the opposite side, over x, over the apothem length. And so if I wanted to solve for x, I could just multiply both sides by x and divide both sides by tangent of 36 degrees, and I would get x is equal to 6.5 divided by tangent.
Tangent, let me do the same color, tangent of 36 degrees. Tangent of 36 degrees. So that's the apothem length, this value right over here. And then what is the perimeter going to be? Well, the perimeter is just going to be, each side is 13 and so it's gonna be 13 times 5, which is equal to, that is equal to 65.
And so our area, our total area, let me write it over here. Our area, I'll just do capital A for area, is going to be 1/2 times the apothem length, which is 6.5, over tangent of 36 degrees times 65. And now let me get out a calculator here. Okay, and let me make sure that I am in degree mode.
Let me just make sure it's, if I were to say the tangent, if I were to say 45 and tangent, yes, it is in degree mode, just making sure. So let me clear things out. So if I have 1/2, 0.5, times, I'll just do the 65, times 65 times 6.5, times 6.5 divided by, divided by 36 degrees, tangent is equal to 290.76.
So I could say that's roughly 291. Well, actually, I'll just go all the way to the hundredths place. So this is going to be approximately 290.76 square units as the area of this regular pentagon.
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