Topic C: Use equations and inequalities to solve geometry problems
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Relating circumference and area
- [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.