About this video
The video finds the area of a regular pentagon whose radius (centre to corner) is 20 cm. It uses a 36-degree angle with cosine to get the apothem 20cos(36°) and sine to get half a side 20sin(36°), so the perimeter is 200sin(36°). The area 1/2 * apothem * perimeter is about 951 square centimetres.
You'll be able to
- tell the difference between the radius and the apothem of a regular polygon.
- use cosine and sine with the radius to find the apothem and half a side length.
- calculate the area of a regular polygon given its radius.
Key words
- radius (of a regular polygon)
- The distance from the centre of the polygon to one of its corners.
- apothem
- The distance from the centre to the middle of a side, meeting the side at a right angle.
- cosine
- In a right triangle, adjacent side divided by hypotenuse.
- sine
- In a right triangle, opposite side divided by hypotenuse.
- SohCahToa
- A memory aid: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Transcript
- [Educator] We're told the following regular pentagon has a radius of 20 centimeters, so that's the radius right over there from the center to one of the corners. And what we wanna do is figure out the area of this regular pentagon. So why don't you pause this video and have a go at it before we do this together.
Okay, now let's do this together. Let's just remind ourselves how we figure out the area of a regular polygon. It's gonna be 1/2 times the apothem length. Apothem length. And I'll remind you what that is. Times the perimeter. And another video I talk about why this makes sense.
It really comes straight out of the fact that you can divide any regular pentagon or any regular polygon into, if in a pentagon's case, you could divide it into five triangles. And so really what you're looking at the area of each of these triangles is gonna be 1/2 height times the base.
And then if you do that five times, and if you were to do all of the math, you essentially can just rewrite that as 1/2 times the apothem length, and then the times the perimeter. But let's actually figure out what's going here because they didn't give us the apothem length, they only gave us the radius.
So we're gonna have to do a little bit of mathematics, probably a little bit of trigonometry to figure out the apothem length. The apothem length is the length of this side right over here, which we do not know. Now we can figure out this angle and we've done this in another video.
Because if we just drew a bunch of apothem's here, if I were to just draw a bunch of aptohem's here, and if you think about rotating around the center, this would be 360 degrees going all the way around. And this is just 1/10 of that. So one 1/10 of 360 degrees is going to be a 36 degree angle.
Now why is that useful? Well, we're trying to figure out, let's just call this x, I don't want to use a, because that'll get confused with area. We're trying to figure out the apothem length, but that's adjacent to this angle. We also know the hypotenuse. So which trig identity involves adjacent and hypotenuse?
Well, cosign does. If we said the cosign of 36 degrees, that is equal to SohCahToa, that's equal to the adjacent over the hypotonus, the adjacent over the hypotonus. In this case, it is of length 20. And so we would get that X is equal to, we just multiply both sides by 20.
We'd get 20 cosine of 36 degrees is equal to x. Now we're not done yet, because we are still going to have trouble figuring out the perimeter. The perimeter is going to be five times each of the side lengths. And so let's do a little bit more trigonometry here.
We know that this is, we know that, well, let me just write it this way. We know what x is now, but how can we figure out what's over here on this length, right over here? Let's call that Y. Well, what we could either use sine or we could use cosine now that we know x.
Let's use sine or sine deals with opposite over hypotonus. So we could say that the sine of 36 degrees is equal to opposite over hypotonus, is equal to Y over 20, or that Y is equal to 20 times a sine of 36 degrees. Sine of 36 degrees. Now why is that useful? Well, Y is exactly half of this, and then we have five of those sides.
So another way to think about it is if we want the entire perimeter, the perimeter's gonna be 10 of these Y's. That's one, two, three, four, five, six, seven, eight, nine, 10. So the perimeter, I'll just call it P, is going to be 10 times this. So that's going to be 200 times the sine of 36 degrees.
So now we have enough to go on. Our area of our pentagon here is going to be 1/2 times the apothem length, 20 sine of 36 degrees times the perimeter times 200. Or let me, that was cosine. The apothem length was coine of 36 degrees. Don't wanna make that mistake.
Cosine of 36 degrees, that's the apothem length right over there. And then I wanna multiply that times 200 sine of 36 degrees, 200 sine of 36 degrees. I don't wanna run outta space. That's the perimeter length. And now I just figure what this is. Let's see this 1/2 times 20 is just going to be 10.
10 times 200 is equal to 2,000. So this is going to be 2,000 times cosine of 36 degrees times sine of 36 degrees, which is going to be equal to. So I have 2,000 times, let me open the parentheses just in case, 36 cosine post parentheses, times open parentheses, 36 sine, close parentheses, is equal to, I'll just round to 951 square centimeters.
So it's approximately equal to 951 square centimeters. And we're done.
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