Relating circumference and area

So we have a circle here, and let's say that we know that its circumference is equal to 6 Pi. I'll write it units, whatever our units happen to be. Let's see if we can figure out, given that its circumference is 6 Pi of these units, what is the area going to be equal to? Pause this video and see if you can figure it out on your own.

First, think about if you could figure out the area for this particular circle. Then, let's see if we can come up with a formula for, given any circle circumference, can we figure out the area and vice versa.

All right, now let's work through this together, and the key here is to realize that from circumference, you can figure out the radius, and then from radius, you can figure out area. So we know that circumference, which is 6 Pi, is equal to 2 Pi times our radius.

So what is the radius going to be? The radius we're talking about, that distance, well, we can divide both sides by 2 Pi. So let's do that. If we divide both sides by 2 Pi to solve for R, what are we left with? Well, we have an R on the right-hand side; we have R is equal to Pi over Pi, that's just 1.

6 divided by 2 is 3, so we get that our radius right over here is equal to 3 units. Then we can use the fact that area is equal to Pi times R squared to figure out the area. This is going to be equal to Pi times 3 squared.

I don't think you have to write parentheses there: Pi times 3 squared, which is, of course, going to be equal to 9 Pi. So for this particular example, when the circumference is 6 Pi units, we're able to figure out that the area is actually going to be 9 Pi square units, or I could write units squared because we're squaring the radius. The radius is three units, so you square that, you get the units squared.

Now let's see if we can come up with a general formula. So we know that circumference is equal to 2 Pi R, and we know that area is equal to Pi R squared. Can we come up with an expression or a formula that relates directly between circumference and area?

I'll give you a hint: solve for R right over here and substitute back into this equation, or vice versa. Pause the video; see if you can do that.

All right, so let's do it over here. Let's solve for R. If we divide both sides by 2 Pi, that’s another color. So if we divide both sides by 2 Pi, and this is exactly what we did up here, what are we left with? We're left with, on the right-hand side, R is equal to C, the circumference, divided by 2 Pi. The radius is equal to the circumference over 2 Pi.

When we're figuring out the area, remember, area is equal to Pi times our radius squared. But we know that our radius could be written as circumference divided by 2 Pi. So instead of radius, I'll write circumference over 2 Pi.

Remember, we want to relate area and circumference. So what is this going to be equal to? We get area is equal to Pi times circumference squared over (2 Pi)^2, which is 4 Pi^2.

Let's see, we have a Pi, or we would have, if we multiply this out, we’d have a Pi in the numerator and two Pis in the denominator being multiplied. So Pi over Pi squared is just 1 over Pi.

And so there you have it: area is equal to circumference squared divided by 4 Pi. Let me write that down. So this is neat; you don't tend to learn this formula, but it's cool that we were able to derive it.

Area is equal to circumference squared over 4 Pi. And we can go the other way around. Given an area, how do we figure out circumference? You could just put the numbers in here, or you could just solve for C.

Let's multiply both sides by 4 Pi. Let's multiply both sides by 4 Pi, and if we do that, what do we get? We would get 4 Pi times the area is equal to our circumference squared.

Then, to solve for the circumference, we just take the square root of both sides. So you would get the square root of 4 Pi times the area is equal to our circumference.

You could simplify this a little bit if you wanted; you could take the four out of the radical. But this is pretty neat how you can relate circumference and area.