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# Circles | Worked example

## Video transcript

- [Narrator] So we have a question here that's dealing with circles. It says, "Ximena is running laps around a circular track. If she runs a total distance of 12 miles in 72 laps, what is the diameter, in miles, of the track? So pause this video and try to work through it on your own before we work through it together. Okay, so let's do it together and so let's visualize what's going on. We have a track that is circular. That's my best freehand drawing of a circle. Imagine an actual perfect circle. And we need to figure out what the diameter of the circle is. So let's call that, that is the diameter of the circle. And they give us some other information. She runs a total of 12 miles in 72 laps. So we should be able to use this to figure out what is the circumference of the circle? The circumference of the circle would be how many miles she runs in one lap. So if we're able to figure out the circumference of the circle, then we can figure out the diameter from that. How do we do that? Well, you might remember that circumference is equal to the number pi times the diameter. Or another way to think about it is, and this is actually where the definition of pie comes from, circumference divided by diameter is defined to be the number pi. Or if you're solving for diameter you divide both sides of this by pi, and you get circumference divided by pi is equal to diameter. So if we can figure out circumference, we just divide by pi and we're going to be able to answer this question. Now what's the circumference here? Well if 12 miles is equal to 72 laps, so 12 miles is equal to 72 laps, how do we figure out what one lap is going to be? Well, we could divide both sides by 72. Because if we divide both sides by 72, so 72 here, we divide 72 over here, on the left hand side 12/72, well that's the same thing as 1/6. 12 divided by 12 is 1, 72 divided by 12 is 6. So we could say 1/6 of a mile, or 1/6 miles, but I'll just say 1/6 of a mile, is equal to 72/72, that's why we divided by 72. It's equal to one laps would not be grammatically correct so I'll say it's equal to one lap. So just like that we've been able to figure out the circumference of the circle. The circumference of the circle is equal to 1/6 of a mile. So what's the diameter? Well it's going to be the circumference divided by pi. So the diameter is going to be the circumference, let me write this, it's a mile, not a meter, so it's going to be 1/6 of a mile, which is the circumference divided by pi. Now that is the same thing as 1/6 pi of a mile. And it's very atypical to one, see circumferences that are fractions in most math problems, although of course in the real world they can be, and this is also a very unusual distance right over here, but that is the diameter of the circumference of that track. So the diameter in miles of the track is 1/6 pi. And we're done.