the circle is arguably the mostfundamental shape in our universe, whether you look atthe shapes of orbits of planets, whether you look atwheels, whether you look at things on kind of amolecular level. the circle just keepsshowing up over and over and over again. so it's probably worthwhile forus to understand some of the properties of the circle.
so the first thing when peoplekind of discovered the circle, and you just have a look at themoon to see a circle, but the first time they said well, whatare the properties of any circle? so the first one they mightwant to say is well, a circle is all of the points that areequal distant from the center of the circle. all of these points along theedge are equal distant from that center right there.
so one of the first thingssomeone might want to ask is what is that distance, thatequal distance that everything is from the center? right there. we call that theradius of the circle. it's just the distance fromthe center out to the edge. if that radius is 3centimeters, then this radius is going to be 3 centimeters. and this radius is goingto be 3 centimeters.
it's never going to change. by definition, a circle is allof the points that are equal distant from the center point. and that distanceis the radius. now the next most interestingthing about that, people might say well, how fatis the circle? how wide is it alongits widest point? or if you just want to cut italong its widest point, what is that distance right there?
and it doesn't have to be justright there, i could have just as easily cut it along itswidest point right there. i just wouldn't be cutting itlike some place like that because that wouldn't bealong its widest point. there's multiple placeswhere i could cut it along its widest point. well, we just saw the radiusand we see that widest point goes through the centerand just keeps going. so it's essentially two radii.
you got one radius thereand then you have another radius over there. we call this distance alongthe widest point of the circle, the diameter. so that is the diameterof the circle. it has a very easyrelationship with the radius. the diameter is equal totwo times the radius. now, the next most interestingthing that you might be wondering about a circle is howfar is it around the circle?
so if you were to get your tapemeasure out and you were to measure around the circle likethat, what's that distance? we call that word thecircumference of the circle. now, we know how the diameterand the radius relates, but how does the circumference relateto, say, the diameter. and if you're not really usedto the diameter, it's very easy to figure out how itrelates to the radius. well, many thousands of yearsago, people took their tape measures out and they keepmeasuring circumferences
and radiuses. and let's say when their tapemeasures weren't so good, let's say they measured thecircumference of the circle and they would get well, itlooks like it's about 3. and then they measure theradius 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 -- letme write this down. so we're worried aboutthe ratio -- let me
write it like this. the ratio of the circumferenceto the diameter. so let's say that somebody hadsome circle over here -- let's say they had this circle, andthe first time with not that good of a tape measure, theymeasured around the circle and they said hey, it'sroughly equal to 3 meters when i go around it. and when i measure thediameter of the circle, it's roughly equal to 1.
ok, that's interesting. maybe the ratio ofthe circumference of the diameter's 3. so maybe the circumferenceis always three times the diameter. well that was just for thiscircle, but let's say they measured some othercircle here. it's like this -- idrew it smaller. let's say that on this circlethey measured around it and
they found out that thecircumference is 6 centimeters, roughly -- we have a badtape measure right then. then they find outthat the diameter is roughly 2 centimeters. and once again, the ratio ofthe circumference of the diameter was roughly 3. ok, this is a neatproperty of circles. maybe the ratio of thecircumference to the diameters always fixed for any circle.
so they said let mestudy this further. so they got bettertape measures. when they got better tapemeasures, they measured hey, my diameter's definitely 1. they say my diameter'sdefinitely 1, but when i measure my circumferencea little bit, i realize it's closer to 3.1. and the same thingwith this over here. they notice that thisratio is closer to 3.1.
then they kept measuring itbetter and better and better, and then they realized thatthey were getting this number, they just kept measuring itbetter and better and they were getting this number 3.14159. and they just kept addingdigits and it would never repeat. it was a strange fascinatingmetaphysical number that kept showing up. so since this number was sofundamental to our universe,
because the circle is sofundamental to our universe, and it just showed upfor every circle. the ratio of the circumferenceof the diameter was this kind of magical number,they gave it a name. they called it pi, or you couldjust give it the latin or the greek letter pi --just like that. that represents this numberwhich is arguably the most fascinating numberin our universe. it first shows up as the ratioof the circumference to the
diameter, but you're going tolearn as you go through your mathematical journey, thatit shows up everywhere. it's one of these fundamentalthings about the universe that just makes you think thatthere's some order to it. but anyway, how can weuse 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 ifyou divide the circumference by
the diameter, you'regoing to get pi. pi is just this number. i could write 3.14159 and justkeep going on and on and on, but that would be a waste ofspace and it would just be hard to deal with, so people justwrite this greek letter pi there. so, how can we relate this? we can multiply both sides ofthis by the diameter and we could say that thecircumference is equal to pi
or since the diameter is equalto 2 times the radius, we could say that the circumference isequal to pi times 2 times the radius. or the form that you'remost likely to see it, it's equal to 2 pi r. so let's see if we can applythat to some problems. so let's say i have a circlejust 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 circumferenceof the circle? the circumference is equal to2 times pi times the radius. so it's going to be equal to 2times pi times the radius, times 3 meters, which isequal 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 goingon 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 youmight want to do it, but for simplicity people just tend toleave our numbers in terms of pi. now i don't know what this isif you multiply 6 times 3.14159, i don't know if youget something close to 19 or 18, maybe it's approximately18 point something
something something. i don't have my calculatorin front of me. but instead of writingthat number, you just write 6 pi there. actually, i think itwouldn't quite cross the threshold to 19 yet. now, let's askanother question. what is the diameterof the circle? well if this radius is 3, thediameter is just twice that.
so it's just going to be 3times 2 or 3 plus 3, which is equal to 6 meters. so the circumference is 6 pimeters, the diameter is 6 meters, the radius is 3 meters. now let's go the other way. let's say i haveanother circle. let's say i haveanother circle here. and i were to tell you thatits circumference is equal to 10 meters -- that's thecircumference of the circle.
if you were to put a tapemeasure to go around it and someone were to ask you what isthe diameter of the circle? well, we know that the diametertimes pi, we know that pi times the diameter is equal tothe circumference; is equal to 10 meters. so to solve for this we wouldjust divide both sides of this equation by pi. the diameter would equal10 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'regoing 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 oftenjust leave it that way. now what is the radius? well, the radius is equalto 1/2 the diameter.
so this whole distance righthere is 10 over pi meters. if we just 1/2 of that, ifwe just want the radius, we just multiply it times 1/2. so you have 1/2 times 10 overpi, which is equal to 1/2 times 10, or you just divide thenumerator and the denominator by 2. you get 5 there, soyou get 5 over pi. so the radius overhere is 5 over pi. nothing super fancy about this.
i think the thing that confusespeople the most is to just realize that pi is a number. pi is just 3.14159 and it justkeeps going on and on and on. there's actually thousands ofbooks written about pi, so it's not like -- i don't knowif there's thousands, i'm exaggerating, but you couldwrite books about this number. but it's just a number. it's a very special number, andif you wanted to write it in a way that you're used to writingnumbers, you could literally
just multiply this out. but most the time people justrealize they like leaving things in terms of pi. anyway, i'll leave you there. in the next video we'll figureout the area of a circle.