Relating diameter and circumference

0:01- [Instructor] We're now gonna talk about what is maybe the most important idea in. We're now going to talk about what is arguably one of the most important ideas in mathematics. So let's take a circle here, and I'm hand drawing it, so it's definitely not going to be a perfect circle, but you can imagine it as a perfect circle.

0:22This is my hand drawn circle. Let's just assume it is a perfect one. And I'm gonna write circle there, for anyone who doesn't think it looks like a circle. And then we could think about the diameter of that circle. And one way you could think about the diameter is if you have the center of the circle right over here.

0:40We have a line segment that goes from one end of the circle through the center to the other side of the circle. So that right over there is the diameter. Now the question that people have thought about 4,000 of years is what's the relationship between the circumference of the circle?

1:04So the circumference is the length going around the circle. What's the relationship between that and the diameter? Let's call the circumference C right over here. Well, you might do something like maybe take a piece of string, the length of this diameter, and see how many times can you fit it around this circumference.

1:26So let's see, you could probably go about, that looks like about one, and then you go about two, and then you go about three. And then there is a little bit more left after that. So it's gonna be three point something diameters, it would take to go around the circle.

1:47Or another way to think about it is if you were to take the circumference and divide it by the diameter, I'll call the diameter D. So the diameter is D here. If you to take that circumference and divided it by that diameter, whatever length that is. It looks like this is going to be approximately equal to three point something.

2:12And people have tried to do this with more and more precise measurements over the millennia, and they actually establish that no matter the circle and no matter the diameter, you're going to get a very special number here that it's going to be 3.14159, and it's just going to keep going and going and going, and it does not repeat.

2:35So it's a very, very, very special number. Now you could imagine, it's oftentimes very useful to think about the relationship between the diameter and a circumference, but it's not very useful to write 3.14159, and you can keep on going. My son memorized 500 digits of this number.

2:53Many people, the world record, I think is memorizing I think 20,000 plus digits. So you can imagine a very important number and a number that you might have heard of before and is likely to make you hungry. So mathematician said, okay, instead of always having to say this number, we are going to call this number pi.

3:12So another way to think about it is the ratio between the circumference and the diameter of a circle is pi. Let me write that again. If you take the circumference, which is going around the circle, divide that by the diameter. This is going to be equal to pi by definition.

3:33And what's really cool, this is on some level a pretty simple thing and we're gonna play with this a little bit more to think about the other ways that you could work with this. But this number pi spelled P-I, no E there. This is used throughout mathematics and shows up all over the place.

3:52In future videos, once you learn calculus, there's this one proof known as the proof of the existence of God. You could debate whether it is or not. But it involves pi as a central figure. But let's think a little bit how this is actually useful. Let's say that we have a circle that has a diameter of 10 centimeters.

4:16Based on that, what is the circumference of that circle going to be? Well, we can go to this right over here, and actually we can manipulate it a bit. If we multiply both sides by the diameter, we get circumference is equal to pi times the diameter, pi times the diameter.

4:37So far diameter is 10 centimeters, together circumference, you just multiply that by pi. So in this case, our circumference will be 10 pi centimeters. Interesting. Now, what if we wanna deal with radii? What's a radius? Well, a radius is half of a diameter. So, for example, that over there is a radius.

5:04Another way to think about it is a diameter is equal to two radii, two radii. So we could put that idea back over here. And so we could also say, that a circumference of a circle is equal to pi. Instead of putting a diameter there, we could say, well, a diameter is the same thing as two radii.

5:28So times 2r, or we could just rewrite this as circumference is equal to, oh, I wrote a pi there again. So my brain is all, 2r, that's a radius right over there. Lemme do that red color not to confuse anyone. Or if we just manipulate things a little bit, you'll typically see this a circumference is equal to 2 pi, 2 times pi times r.

5:58So if someone were to walk up to you on the street and say, alright, I have a circle, a perfect circle. It has a radius of five centimeters. I'll stick with centimeters for now. What is the circumference going to be? Pause this video and think about it. Well, the circumference is gonna be, it's gonna be 2 pi r.

6:21It's gonna be 2 times pi times 5, 5 centimeters in this case, I could keep the units if I want. And so this is gonna be equal to 10 pi centimeters. I know what some of you are thinking. Wait, you're saying the circumference is 10 pi, that doesn't feel like a number.

6:40But remember, pi is a number. It's this long, crazy number that's used throughout mathematics. So if you wanted to approximate what this is, if you multiply 10 times this number, it would be 31.4159, and keep going on centimeters. So it's a little bit less than 31 1/2 centimeters.

7:02This is a number here. It's just easier to write it as 10 pi than to try to write it as this long non-repeating decimal.