0:00:00.780,0:00:04.880 The circle is arguably the most[br]fundamental shape in our 0:00:04.880,0:00:08.490 universe, whether you look at[br]the shapes of orbits of 0:00:08.490,0:00:11.140 planets, whether you look at[br]wheels, whether you look at 0:00:11.140,0:00:12.840 things on kind of a[br]molecular level. 0:00:12.840,0:00:15.860 The circle just keeps[br]showing up over and 0:00:15.860,0:00:17.350 over and over again. 0:00:17.350,0:00:21.110 So it's probably worthwhile for[br]us to understand some of the 0:00:21.110,0:00:23.330 properties of the circle. 0:00:23.330,0:00:26.200 So the first thing when people[br]kind of discovered the circle, 0:00:26.200,0:00:28.960 and you just have a look at the[br]moon to see a circle, but the 0:00:28.960,0:00:31.570 first time they said well, what[br]are the properties 0:00:31.570,0:00:32.910 of any circle? 0:00:32.910,0:00:36.150 So the first one they might[br]want to say is well, a circle 0:00:36.150,0:00:38.690 is all of the points that are[br]equal distant from the 0:00:38.690,0:00:40.440 center of the circle. 0:00:40.440,0:00:43.710 All of these points along the[br]edge are equal distant from 0:00:43.710,0:00:45.210 that center right there. 0:00:45.210,0:00:47.620 So one of the first things[br]someone might want to ask is 0:00:47.620,0:00:50.280 what is that distance, that[br]equal distance that everything 0:00:50.280,0:00:51.770 is from the center? 0:00:51.770,0:00:52.950 Right there. 0:00:52.950,0:00:58.110 We call that the[br]radius of the circle. 0:00:58.110,0:01:00.350 It's just the distance from[br]the center out to the edge. 0:01:00.350,0:01:02.820 If that radius is 3[br]centimeters, then this radius 0:01:02.820,0:01:04.490 is going to be 3 centimeters. 0:01:04.490,0:01:07.170 And this radius is going[br]to be 3 centimeters. 0:01:07.170,0:01:08.270 It's never going to change. 0:01:08.270,0:01:11.690 By definition, a circle is all[br]of the points that are equal 0:01:11.690,0:01:13.400 distant from the center point. 0:01:13.400,0:01:17.050 And that distance[br]is the radius. 0:01:17.050,0:01:19.880 Now the next most interesting[br]thing about that, people might 0:01:19.880,0:01:22.040 say well, how fat[br]is the circle? 0:01:22.040,0:01:26.360 How wide is it along[br]its widest point? 0:01:26.360,0:01:28.710 Or if you just want to cut it[br]along its widest point, what 0:01:28.710,0:01:30.390 is that distance right there? 0:01:30.390,0:01:32.340 And it doesn't have to be just[br]right there, I could have just 0:01:32.340,0:01:35.490 as easily cut it along its[br]widest point right there. 0:01:35.490,0:01:38.520 I just wouldn't be cutting it[br]like some place like that 0:01:38.520,0:01:40.120 because that wouldn't be[br]along its widest point. 0:01:40.120,0:01:41.810 There's multiple places[br]where I could cut it 0:01:41.810,0:01:43.480 along its widest point. 0:01:43.480,0:01:46.730 Well, we just saw the radius[br]and we see that widest point 0:01:46.730,0:01:49.580 goes through the center[br]and just keeps going. 0:01:49.580,0:01:52.920 So it's essentially two radii. 0:01:52.920,0:01:55.640 You got one radius there[br]and then you have another 0:01:55.640,0:01:57.240 radius over there. 0:01:57.240,0:02:01.380 We call this distance along[br]the widest point of the 0:02:01.380,0:02:03.030 circle, the diameter. 0:02:03.030,0:02:06.390 So that is the diameter[br]of the circle. 0:02:06.390,0:02:09.260 It has a very easy[br]relationship with the radius. 0:02:09.260,0:02:16.155 The diameter is equal to[br]two times the radius. 0:02:19.060,0:02:21.790 Now, the next most interesting[br]thing that you might be 0:02:21.790,0:02:24.560 wondering about a circle is how[br]far is it around the circle? 0:02:24.560,0:02:27.340 So if you were to get your tape[br]measure out and you were to 0:02:27.340,0:02:35.910 measure around the circle like[br]that, what's that distance? 0:02:35.910,0:02:44.710 We call that word the[br]circumference of the circle. 0:02:44.710,0:02:47.440 Now, we know how the diameter[br]and the radius relates, but how 0:02:47.440,0:02:49.790 does the circumference relate[br]to, say, the diameter. 0:02:49.790,0:02:51.550 And if you're not really used[br]to the diameter, it's very 0:02:51.550,0:02:54.290 easy to figure out how it[br]relates to the radius. 0:02:54.290,0:02:57.130 Well, many thousands of years[br]ago, people took their tape 0:02:57.130,0:02:58.890 measures out and they keep[br]measuring circumferences 0:02:58.890,0:03:00.430 and radiuses. 0:03:00.430,0:03:03.280 And let's say when their tape[br]measures weren't so good, 0:03:03.280,0:03:05.010 let's say they measured the[br]circumference of the circle 0:03:05.010,0:03:07.960 and they would get well, it[br]looks like it's about 3. 0:03:07.960,0:03:11.600 And then they measure the[br]radius of the circle right here 0:03:11.600,0:03:14.280 or the diameter of that circle,[br]and they'd say oh, the diameter 0:03:14.280,0:03:16.290 looks like it's about 1. 0:03:16.290,0:03:17.740 So they would say -- let[br]me write this down. 0:03:17.740,0:03:21.750 So we're worried about[br]the ratio -- let me 0:03:21.750,0:03:22.660 write it like this. 0:03:22.660,0:03:33.955 The ratio of the circumference[br]to the diameter. 0:03:37.560,0:03:40.900 So let's say that somebody had[br]some circle over here -- let's 0:03:40.900,0:03:43.170 say they had this circle, and[br]the first time with not that 0:03:43.170,0:03:45.880 good of a tape measure, they[br]measured around the circle 0:03:45.880,0:03:49.340 and they said hey, it's[br]roughly equal to 3 meters 0:03:49.340,0:03:50.490 when I go around it. 0:03:50.490,0:03:52.800 And when I measure the[br]diameter of the circle, 0:03:52.800,0:03:55.050 it's roughly equal to 1. 0:03:55.050,0:03:56.000 OK, that's interesting. 0:03:56.000,0:03:57.520 Maybe the ratio of[br]the circumference of 0:03:57.520,0:03:58.500 the diameter's 3. 0:03:58.500,0:04:00.820 So maybe the circumference[br]is always three 0:04:00.820,0:04:02.020 times the diameter. 0:04:02.020,0:04:03.610 Well that was just for this[br]circle, but let's say they 0:04:03.610,0:04:05.720 measured some other[br]circle here. 0:04:05.720,0:04:07.870 It's like this -- I[br]drew it smaller. 0:04:07.870,0:04:11.200 Let's say that on this circle[br]they measured around it and 0:04:11.200,0:04:14.960 they found out that the[br]circumference is 6 centimeters, 0:04:14.960,0:04:18.210 roughly -- we have a bad[br]tape measure right then. 0:04:18.210,0:04:21.710 Then they find out[br]that the diameter is 0:04:21.710,0:04:23.520 roughly 2 centimeters. 0:04:23.520,0:04:25.490 And once again, the ratio of[br]the circumference of the 0:04:25.490,0:04:30.230 diameter was roughly 3. 0:04:30.230,0:04:32.140 OK, this is a neat[br]property of circles. 0:04:32.140,0:04:35.430 Maybe the ratio of the[br]circumference to the diameters 0:04:35.430,0:04:38.080 always fixed for any circle. 0:04:38.080,0:04:40.260 So they said let me[br]study this further. 0:04:40.260,0:04:42.510 So they got better[br]tape measures. 0:04:42.510,0:04:45.090 When they got better tape[br]measures, they measured hey, 0:04:45.090,0:04:47.630 my diameter's definitely 1. 0:04:47.630,0:04:49.430 They say my diameter's[br]definitely 1, but when I 0:04:49.430,0:04:51.810 measure my circumference[br]a little bit, I realize 0:04:51.810,0:04:53.040 it's closer to 3.1. 0:04:56.000,0:04:57.290 And the same thing[br]with this over here. 0:04:57.290,0:04:59.370 They notice that this[br]ratio is closer to 3.1. 0:04:59.370,0:05:01.830 Then they kept measuring it[br]better and better and better, 0:05:01.830,0:05:05.200 and then they realized that[br]they were getting this number, 0:05:05.200,0:05:07.300 they just kept measuring it[br]better and better and they were 0:05:07.300,0:05:10.850 getting this number 3.14159. 0:05:10.850,0:05:12.550 And they just kept adding[br]digits and it would 0:05:12.550,0:05:13.620 never repeat. 0:05:13.620,0:05:16.640 It was a strange fascinating[br]metaphysical number 0:05:16.640,0:05:18.300 that kept showing up. 0:05:18.300,0:05:20.940 So since this number was so[br]fundamental to our universe, 0:05:20.940,0:05:23.500 because the circle is so[br]fundamental to our universe, 0:05:23.500,0:05:26.680 and it just showed up[br]for every circle. 0:05:26.680,0:05:28.865 The ratio of the circumference[br]of the diameter was this 0:05:28.865,0:05:32.390 kind of magical number,[br]they gave it a name. 0:05:32.390,0:05:37.580 They called it pi, or you could[br]just give it the Latin or the 0:05:37.580,0:05:41.880 Greek letter pi --[br]just like that. 0:05:41.880,0:05:45.090 That represents this number[br]which is arguably the most 0:05:45.090,0:05:46.790 fascinating number[br]in our universe. 0:05:46.790,0:05:50.430 It first shows up as the ratio[br]of the circumference to the 0:05:50.430,0:05:54.070 diameter, but you're going to[br]learn as you go through your 0:05:54.070,0:05:57.160 mathematical journey, that[br]it shows up everywhere. 0:05:57.160,0:05:59.500 It's one of these fundamental[br]things about the universe that 0:05:59.500,0:06:03.060 just makes you think that[br]there's some order to it. 0:06:03.060,0:06:07.750 But anyway, how can we[br]use this in I guess 0:06:07.750,0:06:09.330 our basic mathematics? 0:06:09.330,0:06:12.490 So we know, or I'm telling you,[br]that the ratio of the 0:06:12.490,0:06:19.420 circumference to the diameter[br]-- when I say the ratio, 0:06:19.420,0:06:21.390 literally I'm just saying if[br]you divide the circumference by 0:06:21.390,0:06:28.400 the diameter, you're[br]going to get pi. 0:06:28.400,0:06:29.500 Pi is just this number. 0:06:29.500,0:06:33.570 I could write 3.14159 and just[br]keep going on and on and on, 0:06:33.570,0:06:35.950 but that would be a waste of[br]space and it would just be hard 0:06:35.950,0:06:38.570 to deal with, so people just[br]write this Greek 0:06:38.570,0:06:40.330 letter pi there. 0:06:40.330,0:06:41.850 So, how can we relate this? 0:06:41.850,0:06:44.920 We can multiply both sides of[br]this by the diameter and we 0:06:44.920,0:06:48.640 could say that the[br]circumference is equal to pi 0:06:48.640,0:06:50.820 times the diameter. 0:06:50.820,0:06:55.570 Or since the diameter is equal[br]to 2 times the radius, we could 0:06:55.570,0:06:59.420 say that the circumference is[br]equal to pi times 2 0:06:59.420,0:07:00.360 times the radius. 0:07:00.360,0:07:03.450 Or the form that you're[br]most likely to see it, 0:07:03.450,0:07:07.360 it's equal to 2 pi r. 0:07:07.360,0:07:11.220 So let's see if we can apply[br]that to some problems. 0:07:11.220,0:07:17.240 So let's say I have a circle[br]just like that, and I were to 0:07:17.240,0:07:22.600 tell you it has a radius --[br]it's radius right there is 3. 0:07:22.600,0:07:28.820 So, 3 -- let me write this down[br]-- so the radius is equal to 3. 0:07:28.820,0:07:32.310 Maybe it's 3 meters --[br]put some units in there. 0:07:32.310,0:07:34.660 What is the circumference[br]of the circle? 0:07:34.660,0:07:38.180 The circumference is equal to[br]2 times pi times the radius. 0:07:38.180,0:07:42.090 So it's going to be equal to 2[br]times pi times the radius, 0:07:42.090,0:07:47.280 times 3 meters, which is[br]equal to 6 meters times 0:07:47.280,0:07:49.520 pi or 6 pi meters. 0:07:49.520,0:07:52.430 6 pi meters. 0:07:52.430,0:07:53.740 Now I could multiply this out. 0:07:53.740,0:07:55.900 Remember pi is just a number. 0:07:55.900,0:07:59.680 Pi is 3.14159 going[br]on and on and on. 0:07:59.680,0:08:03.460 So if I multiply 6 times that,[br]maybe I'll get 18 point 0:08:03.460,0:08:05.600 something something something. 0:08:05.600,0:08:07.850 If you have your calculator you[br]might want to do it, but for 0:08:07.850,0:08:10.490 simplicity people just tend to[br]leave our numbers 0:08:10.490,0:08:12.120 in terms of pi. 0:08:12.120,0:08:14.020 Now I don't know what this is[br]if you multiply 6 times 0:08:14.020,0:08:18.510 3.14159, I don't know if you[br]get something close to 19 or 0:08:18.510,0:08:20.910 18, maybe it's approximately[br]18 point something 0:08:20.910,0:08:21.720 something something. 0:08:21.720,0:08:23.450 I don't have my calculator[br]in front of me. 0:08:23.450,0:08:25.300 But instead of writing[br]that number, you just 0:08:25.300,0:08:27.060 write 6 pi there. 0:08:27.060,0:08:29.770 Actually, I think it[br]wouldn't quite cross the 0:08:29.770,0:08:31.430 threshold to 19 yet. 0:08:31.430,0:08:33.770 Now, let's ask[br]another question. 0:08:33.770,0:08:35.270 What is the diameter[br]of the circle? 0:08:38.580,0:08:42.690 Well if this radius is 3, the[br]diameter is just twice that. 0:08:42.690,0:08:45.730 So it's just going to be 3[br]times 2 or 3 plus 3, which 0:08:45.730,0:08:47.170 is equal to 6 meters. 0:08:47.170,0:08:50.750 So the circumference is 6 pi[br]meters, the diameter is 6 0:08:50.750,0:08:53.620 meters, the radius is 3 meters. 0:08:53.620,0:08:55.110 Now let's go the other way. 0:08:55.110,0:08:57.310 Let's say I have[br]another circle. 0:08:57.310,0:09:01.220 Let's say I have[br]another circle here. 0:09:01.220,0:09:04.620 And I were to tell you that[br]its circumference is equal 0:09:04.620,0:09:08.560 to 10 meters -- that's the[br]circumference of the circle. 0:09:08.560,0:09:10.990 If you were to put a tape[br]measure to go around it and 0:09:10.990,0:09:18.370 someone were to ask you what is[br]the diameter of the circle? 0:09:18.370,0:09:22.810 Well, we know that the diameter[br]times pi, we know that pi times 0:09:22.810,0:09:26.830 the diameter is equal to[br]the circumference; is 0:09:26.830,0:09:28.700 equal to 10 meters. 0:09:28.700,0:09:31.020 So to solve for this we would[br]just divide both sides 0:09:31.020,0:09:32.520 of this equation by pi. 0:09:32.520,0:09:35.860 The diameter would equal[br]10 meters over pi or 0:09:35.860,0:09:38.710 10 over pi meters. 0:09:38.710,0:09:40.020 And that is just a number. 0:09:40.020,0:09:42.540 If you have your calculator,[br]you could actually divide 10 0:09:42.540,0:09:46.030 divided by 3.14159, you're[br]going to get 3 point something 0:09:46.030,0:09:47.500 something something meters. 0:09:47.500,0:09:48.960 I can't do it in my head. 0:09:48.960,0:09:50.070 But this is just a number. 0:09:50.070,0:09:53.320 But for simplicity we often[br]just leave it that way. 0:09:53.320,0:09:55.270 Now what is the radius? 0:09:55.270,0:09:58.590 Well, the radius is equal[br]to 1/2 the diameter. 0:09:58.590,0:10:02.870 So this whole distance right[br]here is 10 over pi meters. 0:10:02.870,0:10:06.230 If we just 1/2 of that, if[br]we just want the radius, we 0:10:06.230,0:10:07.580 just multiply it times 1/2. 0:10:07.580,0:10:13.160 So you have 1/2 times 10 over[br]pi, which is equal to 1/2 times 0:10:13.160,0:10:16.770 10, or you just divide the[br]numerator and the 0:10:16.770,0:10:18.140 denominator by 2. 0:10:18.140,0:10:21.130 You get 5 there, so[br]you get 5 over pi. 0:10:21.130,0:10:23.890 So the radius over[br]here is 5 over pi. 0:10:23.890,0:10:25.690 Nothing super fancy about this. 0:10:25.690,0:10:29.760 I think the thing that confuses[br]people the most is to just 0:10:29.760,0:10:31.820 realize that pi is a number. 0:10:31.820,0:10:38.640 Pi is just 3.14159 and it just[br]keeps going on and on and on. 0:10:38.640,0:10:41.950 There's actually thousands of[br]books written about pi, so 0:10:41.950,0:10:45.100 it's not like -- I don't know[br]if there's thousands, I'm 0:10:45.100,0:10:48.340 exaggerating, but you could[br]write books about this number. 0:10:48.340,0:10:49.340 But it's just a number. 0:10:49.340,0:10:52.480 It's a very special number, and[br]if you wanted to write it in a 0:10:52.480,0:10:54.390 way that you're used to writing[br]numbers, you could literally 0:10:54.390,0:10:55.680 just multiply this out. 0:10:55.680,0:10:58.530 But most the time people just[br]realize they like leaving 0:10:58.530,0:11:00.640 things in terms of pi. 0:11:00.640,0:11:01.680 Anyway, I'll leave you there. 0:11:01.680,0:11:05.090 In the next video we'll figure[br]out the area of a circle.