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## Topic C: Use equations and inequalities to solve geometry problems

Current time:0:00Total duration:4:51

# Relating circumference and area

## Video transcript

- [Instructor] So we have a circle here, let's say that we know
that its circumference is equal to six 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. And first, think about
if you could figure out the area for this particular circle, and then let's see if we can come up with a formula for
given any circumference, can we figure out the area and vice versa. Alright now let's work
through this together. The key here is to realize that from circumference
you can figure out radius and then from radius
you can figure out area. So we know that circumference,
which is six pi, so we know six pi is equal
to two pi times r, radius. And so what is the radius going to be? So the radius, we're
talking about that distance, well we can divide both sides by two pi, so let's do that. If we divide both sides by two 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 divided by pi, that's just one. Six divided by two is three. So we get that our
radius, right over here, is equal to three units. And then we can use the fact that are is equal to pi, r squared, to figure out the area. This is going to be equal
to pi times three squared. Oh and you have to
write parenthesis there. Pi, times three, squared,
which is of course going to be equal to nine pi. So for this particular example, when the circumference is six pi units, we're able to figure out that the area, this is actually going to
be nine pi square units, or I could write units squared. Cause 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 two 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? And I'll give you a hint, solve for, you could solve for r right over here, and substitute back into
this equation, or vice versa. Pause the video, see if you can do that. Alright, so let's do it over here. Let's solve for r. If we divide both sides by two pi, do it in another color, so if we divide both sides by two 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 two pi, the radius is equal to the
circumference over two pi. And so when we're figuring out the area, area, remember, is equal to
pi times our radius squared. But we know that our
radius could be written as circumference divided by two pi. So instead of radius I'll write
circumference over two pi. Remember, we want to relate between area and circumference. And so what is this going to be equal to? We get area is equal to pi
times circumference squared, over two pi squared is four pi squared. Now let's see, we have a pi, if we multiplied this out,
we'd have a pi in the numerator and a pi in the denominator, or two pi's in the
denominator being multiplied. So pi divided by pi squared, is just one over pi, and
so there you have it. Area is equal to circumference
squared divided by four pi. Let me write that down. So this is a 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 four pi. And we can go the other way around. Given an area, how do we
figure out circumference? Well you could just put
the numbers in here, or you could just solve for c. Let's multiply both sides by four pi. Let's multiply both sides by four pi, and if we do that, what do we get? We would get four pi
times the area is equal to our circumference squared and then to solve for the circumference we just take the square
root of both sides. So you would get the square
root of four pi times the area is equal to r circumference. And 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.