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Current time:0:00Total duration:11:05

Video transcript

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 know you just have to look at the moon to see a circle but the first time they say well you know what 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 equidistant from the center of the circle right 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 the radius of the circle is just the distance from the center out to the edge if that radius is 3 centimeters and 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 equidistant from the center point and that distant is that distance is the radius now the next most interesting thing about that people might say well how fat is the circle how how 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 someplace 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 this keeps going so it's essentially two radii you get one radius there and then you have another radius over there and we call this distance along the widest point on the circle the diameter so that is the diameter of the circle it has a very easy relationship with the radius the diameter diameter is equal to two times two times the radius 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 around the circle like that if you were to measure around the circle like that what's that distance we call that word the circumference circumference circumference of the circle now we know how the diameter and the radius relates but how does the circumference relate to say the diameter if we not release it a diameter it's very easy to figure out how to release it to the radius well many thousands of years ago people took their tape measures out and they keep measuring circumference is and radiuses and let's say when their tape measures weren't so good they you know let's say they measure the circumference of the circle and they would get it well it looks like it's about three and then they measure the radius of this circle right here and they are the diameter of that circle and they say well that diameter looks like it's about one so they would say let me write this down so we're worried about we're worried about the ratio where we write like this the ratio of the circumference circumference to the diameter to the diameter so let's say that you know 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 three meters when I go around it and when I measure the diameter of the circle it's roughly equal to one okay that's interesting it may be the ratio of the circumference to the diameter is three so maybe the circumference is always three times the diameter well that was just for this circle but they don't say they measured some other circle here it sits like this I drew it smaller but 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 and then they find out that the diameter the diameter is roughly 2 centimeters and once again the ratio of the cumference the diameter circumference to diameter was roughly three okay this is a neat property of circles maybe the ratio of the circumference to the diameter is 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 you know my diameter is definitely one they say my diameter is definitely one but when I measure my circumference a little bit I realize it's more it's closer to three point one closer to three point one and the same thing with this over here they notice that this ratio is closer to three point one and then they kept measuring it better and better and better and then they realize that it was they were getting this number it just kept measuring it better and better and better and they were getting this number 3.14159 and they just kept adding digits and it would never repeat and it was a strange fascinating metaphysical number that kept showing up and so since this number was so fundamental to our universe because the circle is so fundamental to our universe and it just showed up it just showed up for every circle the ratio of the circumference diameter was this kind of magical number they gave it a nut 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 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 some order to it but anyway how can we use this in our in our in our in our just I guess our basic mathematics so we know we're I'm telling you that the ratio of the circumference circumference circumference to the diameter when I say the ratio at luery I'm just saying if you divide the circumference by the diameter but 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 set by the diameter and we could say that the circumference is equal to PI times the diameter or since the diameter is equal to two times the radius we could say that the circumference is equal to PI times two times the radius or the form that you're most likely to see it it's equal to two PI R so let's see if we can apply that to some problems so let's say I have a circle I have a circle just like that and I were to tell you it has a radius its radius right there is three so three let me write this down so the radius is equal to three maybe it's three meters but some units in there what is the circumference of the circle their circumference is equal to two times pi times the radius is going to be equal to two times pi times the radius times times three meters which is equal to six meters times pi or six pi meters six PI meters now I could multiply this out remember pi is just a number pi is 3.14159 i'm going on and on and on so if I multiply six times that maybe I'll get you know 18 point something something something I'll get some 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 not to know what this is if you multiply 6 times 3.14159 if you get something close to 19 or 18 maybe it's approximately 18 points 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 yeah I think it would it wouldn't quite cross the threshold to 19 yet now what if I were to go do it well let's ask another question what is the diameter of the circle what is the diameter 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 three 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 the 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 what is the diameter of the circle well we know its 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 so 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 gonna get three points 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 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 want half of that if we just want the radius we just multiply it times half so you have 1/2 times 10 over pi which is equal to 1/2 times 10 or you could just say this to divide the numerator in the denominator by 2 you know 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 it just keeps going on and on and on you could there's you could write there's there's actually thousands of books written about pi so it's not like you know and then I don't know if there's thousands but maybe 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 can literally just multiply this out but most of 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