Sal finds the radius and diameter of a circle given the circumference.
- [Voiceover] Let's say that we know that the circumference of a circle is 49 pi. Based on that, let's see if we can figure out what the radius of that same circle is going to be. And I encourage you, and I'll write equals here. And I encourage you to pause the video, and see if you can figure it out on your own. Let's just draw the circle to help visualize it. I'll just do a hand-drawn circle, clearly not a perfect circle right over here. We know that if its radius is of length r, that the circumference is going to be two pi times r. So, I could write the circumference is equal to two pi times r. In fact, the number pi, the standard definition for it, is just the ratio between the circumference and the diameter of a circle. Now, why is that? Well, if the diameter here is two r, right? We have r and then have another r. We see that the circumference is pi times two r, or we can say that the ratio between the circumference and the diameter, which is the ratio between c and two r, that's just going to be pi. Anyway, I've gone on longer than I need to just to solve this problem. We can go to this original formula here, saying the circumference is two pi times r, and we can just substitute in 49 pi for the circumference. So, we could say 49 pi is going to be equal to two pi times the radius. Now, let's see, we can divide both sides by two pi to solve for r. So, dividing both sides by two pi. On the right-hand side, the two pis cancel out. On the left-hand side, pi divided by pi cancels out. 49 divided by two is 24.5. So, if the circumference is 49 pi whatever units, then the radius is going to be 24.5 of those units. Let's do one more of these. Let's say that we have a circle whose circumference, I'll just say C, is equal to 1600 pi. My question is what is the diameter? The diameter of the circle is equal to what? Just as we said that the circumference could be written as two pi r or as pi times two r, two r is just the diameter. So, we could say that the circumference is equal to pi times the diameter. Once again, that comes out of that traditional definition of pi as the ratio between the circumference and the diameter. You could say that the ratio between the circumference and the diameter is equal to pi. Circles are this very fundamental thing in the universe, and you take the ratio of the circumference and the diameter, you get this magical and mystical number that we see that keeps popping up in mathematics. Anyway, back to the problem. If we say the circumference is 1600 pi, and this is equal to pi times the diameter, we can just divide both sides by pi to get the diameter, which is going to be 1600. The circumference is 1600 pi units, whatever units those are, maybe meters. Then, the diameter is just going to be 1600 of those units, or in this case, maybe meters. And we're all done.