Area and circumference of circles
The circle is arguably the most fundamental shape in our universe, whether you look at the shapes of orbits of planets, whether you look at wheels, whether you look at things on kind of a molecular level. The circle just keeps showing up over and over and over again. So it's probably worthwhile for us to understand some of the properties of the circle. So the first thing when people kind of discovered the circle, and you just have a look at the moon to see a circle, but the first time they said well, what are the properties of any circle? So the first one they might want to say is well, a circle is all of the points that are equal distant from the center of the circle. All of these points along the edge are equal distant from that center right there. So one of the first things someone might want to ask is what is that distance, that equal distance that everything is from the center? Right there. We call that the radius of the circle. It's just the distance from the center out to the edge. If that radius is 3 centimeters, then this radius is going to be 3 centimeters. And this radius is going to be 3 centimeters. It's never going to change. By definition, a circle is all of the points that are equal distant from the center point. And that distance is the radius. Now the next most interesting thing about that, people might say well, how fat is the circle? How wide is it along its widest point? Or if you just want to cut it along its widest point, what is that distance right there? And it doesn't have to be just right there, I could have just as easily cut it along its widest point right there. I just wouldn't be cutting it like some place like that because that wouldn't be along its widest point. There's multiple places where I could cut it along its widest point. Well, we just saw the radius and we see that widest point goes through the center and just keeps going. So it's essentially two radii. You got one radius there and then you have another radius over there. We call this distance along the widest point of the circle, the diameter. So that is the diameter of the circle. It has a very easy relationship with the radius. The diameter is equal to two times the radius. Now, the next most interesting thing that you might be wondering about a circle is how far is it around the circle? So if you were to get your tape measure out and you were to measure around the circle like that, what's that distance? We call that word the circumference of the circle. Now, we know how the diameter and the radius relates, but how does the circumference relate to, say, the diameter. And if you're not really used to the diameter, it's very easy to figure out how it relates to the radius. Well, many thousands of years ago, people took their tape measures out and they keep measuring circumferences and radiuses. And let's say when their tape measures weren't so good, let's say they measured the circumference of the circle and they would get well, it looks like it's about 3. And then they measure the radius of the circle right here or the diameter of that circle, and they'd say oh, the diameter looks like it's about 1. So they would say -- let me write this down. So we're worried about the ratio -- let me write it like this. The ratio of the circumference to the diameter. So let's say that somebody had some circle over here -- let's say they had this circle, and the first time with not that good of a tape measure, they measured around the circle and they said hey, it's roughly equal to 3 meters when I go around it. And when I measure the diameter of the circle, it's roughly equal to 1. OK, that's interesting. Maybe the ratio of the circumference of the diameter's 3. So maybe the circumference is always three times the diameter. Well that was just for this circle, but let's say they measured some other circle here. It's like this -- I drew it smaller. Let's say that on this circle they measured around it and they found out that the circumference is 6 centimeters, roughly -- we have a bad tape measure right then. Then they find out that the diameter is roughly 2 centimeters. And once again, the ratio of the circumference of the diameter was roughly 3. OK, this is a neat property of circles. Maybe the ratio of the circumference to the diameters always fixed for any circle. So they said let me study this further. So they got better tape measures. When they got better tape measures, they measured hey, my diameter's definitely 1. They say my diameter's definitely 1, but when I measure my circumference a little bit, I realize it's closer to 3.1. And the same thing with this over here. They notice that this ratio is closer to 3.1. Then they kept measuring it better and better and better, and then they realized that they were getting this number, they just kept measuring it better and better and they were getting this number 3.14159. And they just kept adding digits and it would never repeat. It was a strange fascinating metaphysical number that kept showing up. So since this number was so fundamental to our universe, because the circle is so fundamental to our universe, and it just showed up for every circle. The ratio of the circumference of the diameter was this kind of magical number, they gave it a name. They called it pi, or you could just give it the Latin or the Greek letter pi -- just like that. That represents this number which is arguably the most fascinating number in our universe. It first shows up as the ratio of the circumference to the diameter, but you're going to learn as you go through your mathematical journey, that it shows up everywhere. It's one of these fundamental things about the universe that just makes you think that there's some order to it. But anyway, how can we use this in I guess our basic mathematics? So we know, or I'm telling you, that the ratio of the circumference to the diameter -- when I say the ratio, literally I'm just saying if you divide the circumference by the diameter, you're going to get pi. Pi is just this number. I could write 3.14159 and just keep going on and on and on, but that would be a waste of space and it would just be hard to deal with, so people just write this Greek letter pi there. So, how can we relate this? We can multiply both sides of this by the diameter and we could say that the circumference is equal to pi times the diameter. Or since the diameter is equal to 2 times the radius, we could say that the circumference is equal to pi times 2 times the radius. Or the form that you're most likely to see it, it's equal to 2 pi r. So let's see if we can apply that to some problems. So let's say I have a circle just like that, and I were to tell you it has a radius -- it's radius right there is 3. So, 3 -- let me write this down -- so the radius is equal to 3. Maybe it's 3 meters -- put some units in there. What is the circumference of the circle? The circumference is equal to 2 times 5 times the radius. So it's going to be equal to 2 times pi times the radius, times 3 meters, which is equal to 6 meters times pi or 6 pi meters. 6 pi meters. Now I could multiply this out. Remember pi is just a number. Pi is 3.14159 going on and on and on. So if I multiply 6 times that, maybe I'll get 18 point something something something. If you have your calculator you might want to do it, but for simplicity people just tend to leave our numbers in terms of pi. Now I don't know what this is if you multiply 6 times 3.14159, I don't know if you get something close to 19 or 18, maybe it's approximately 18 point something something something. I don't have my calculator in front of me. But instead of writing that number, you just write 6 pi there. Actually, I think it wouldn't quite cross the threshold to 19 yet. Now, let's ask another question. What is the diameter of the circle? Well if this radius is 3, the diameter is just twice that. So it's just going to be 3 times 2 or 3 plus 3, which is equal to 6 meters. So the circumference is 6 pi meters, the diameter is 6 meters, the radius is 3 meters. Now let's go the other way. Let's say I have another circle. Let's say I have another circle here. And I were to tell you that its circumference is equal to 10 meters -- that's the circumference of the circle. If you were to put a tape measure to go around it and someone were to ask you what is the diameter of the circle? Well, we know that the diameter times pi, we know that pi times the diameter is equal to the circumference; is equal to 10 meters. So to solve for this we would just divide both sides of this equation by pi. The diameter would equal 10 meters over pi or 10 over pi meters. And that is just a number. If you have your calculator, you could actually divide 10 divided by 3.14159, you're going to get 3 point something something something meters. I can't do it in my head. But this is just a number. But for simplicity we often just leave it that way. Now what is the radius? Well, the radius is equal to 1/2 the diameter. So this whole distance right here is 10 over pi meters. If we just 1/2 of that, if we just want the radius, we just multiply it times 1/2. So you have 1/2 times 10 over pi, which is equal to 1/2 times 10, or you just divide the numerator and the denominator by 2. You get 5 there, so you get 5 over pi. So the radius over here is 5 over pi. Nothing super fancy about this. I think the thing that confuses people the most is to just realize that pi is a number. Pi is just 3.14159 and it just keeps going on and on and on. There's actually thousands of books written about pi, so it's not like -- I don't know if there's thousands, I'm exaggerating, but you could write books about this number. But it's just a number. It's a very special number, and if you wanted to write it in a way that you're used to writing numbers, you could literally just multiply this out. But most the time people just realize they like leaving things in terms of pi. Anyway, I'll leave you there. In the next video we'll figure out the area of a circle.