Area and circumference challenge problems
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If we know some circle has an area of 36pi-- so it has an area of 36pi-- can we figure out what the circumference of this circle is? And I encourage you to pause this video, and try to think about that question. Well, from the area, we could figure out what the radius is, and then from that radius, we can figure out what its circumference is. So we know that the area, which is 36pi, is equal to pi r squared. And so if you look at it on both sides of this equation, if we divide-- let me rewrite it so it's a little bit clearer in a different color. So we could set up an equation pi r squared is equal to 36pi. Now, if we want to solve for the radius the first thing that we might want to do is divide both sides by pi. Then, we're left with r squared is equal to 36. Now, if we just solve this as a pure math equation, you might say, OK, we could take the positive and negative square root of 36. r could be plus or minus 6, but we need to remember that r is a distance, so we only care about the positive. So if we take the principal root of 36, we get r is equal to 6. From there, we can use this to figure out the circumference. So the circumference is equal to 2 pi r. Circumference is equal to 2 pi r. And in this case, r is equal to 6. So it's equal to 2 pi times 6, which is going to be equal to 12pi. So that's straightforward, area 36pi, we leverage pi r squared to figure out that the radius was 6, and then from that we were able to figure out that the circumference was 12pi.