[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.69,0:00:05.61,Default,,0000,0000,0000,,So we've got a circle here--\Ndoesn't look like a perfect
Dialogue: 0,0:00:05.61,0:00:09.69,Default,,0000,0000,0000,,circle, but we can use our\Nimaginations --and let's say
Dialogue: 0,0:00:09.69,0:00:21.78,Default,,0000,0000,0000,,it's got a radius of 3 meters.
Dialogue: 0,0:00:21.78,0:00:24.58,Default,,0000,0000,0000,,My question, or the question\Nwe're going to answer in this
Dialogue: 0,0:00:24.58,0:00:29.02,Default,,0000,0000,0000,,video is what is the\Narea of this circle?
Dialogue: 0,0:00:29.02,0:00:31.90,Default,,0000,0000,0000,,And remember, the area is just\Nhow much space this circle
Dialogue: 0,0:00:31.90,0:00:34.76,Default,,0000,0000,0000,,takes up on a surface, or on\Nthis computer screen that
Dialogue: 0,0:00:34.76,0:00:36.28,Default,,0000,0000,0000,,you're watching, or on\Nthis piece of paper.
Dialogue: 0,0:00:36.28,0:00:39.33,Default,,0000,0000,0000,,
Dialogue: 0,0:00:39.33,0:00:42.10,Default,,0000,0000,0000,,If this was a room, it's how\Nmuch carpeting you would need
Dialogue: 0,0:00:42.10,0:00:44.24,Default,,0000,0000,0000,,to fill out this circular room.
Dialogue: 0,0:00:44.24,0:00:45.80,Default,,0000,0000,0000,,That's what the area is.
Dialogue: 0,0:00:45.80,0:00:48.86,Default,,0000,0000,0000,,
Dialogue: 0,0:00:48.86,0:00:51.37,Default,,0000,0000,0000,,Now, I'm not going to prove it\Nto you, and we'll do that
Dialogue: 0,0:00:51.37,0:00:54.95,Default,,0000,0000,0000,,later, but the area for circle\Njust takes on a fairly
Dialogue: 0,0:00:54.95,0:00:58.52,Default,,0000,0000,0000,,straightforward formula and I\Nwant to just get you used to
Dialogue: 0,0:00:58.52,0:00:59.61,Default,,0000,0000,0000,,applying that formula.
Dialogue: 0,0:00:59.61,0:01:04.86,Default,,0000,0000,0000,,So the area of a circle\Nis equal to pi.
Dialogue: 0,0:01:04.86,0:01:08.71,Default,,0000,0000,0000,,Remember, pi was that number\Nthat people figured out was the
Dialogue: 0,0:01:08.71,0:01:11.75,Default,,0000,0000,0000,,ratio between the circumference\Nand the diameter of the circle.
Dialogue: 0,0:01:11.75,0:01:15.42,Default,,0000,0000,0000,,It's 3.14159, keeps\Ngoing on and on and on.
Dialogue: 0,0:01:15.42,0:01:17.88,Default,,0000,0000,0000,,It's just the number, but\Nit's a very magical number.
Dialogue: 0,0:01:17.88,0:01:19.76,Default,,0000,0000,0000,,Pi times the radius squared.
Dialogue: 0,0:01:19.76,0:01:22.60,Default,,0000,0000,0000,,
Dialogue: 0,0:01:22.60,0:01:26.81,Default,,0000,0000,0000,,In fact another way of defining\Npi:-- you could even rewrite
Dialogue: 0,0:01:26.81,0:01:32.69,Default,,0000,0000,0000,,this right here --the area over\Nyour radius squared- so
Dialogue: 0,0:01:32.69,0:01:33.65,Default,,0000,0000,0000,,this is your radius.
Dialogue: 0,0:01:33.65,0:01:37.11,Default,,0000,0000,0000,,If you multiply the radius\Ntimes itself you could imagine
Dialogue: 0,0:01:37.11,0:01:41.10,Default,,0000,0000,0000,,that would be the area of a\Ncube that's like that --that
Dialogue: 0,0:01:41.10,0:01:44.18,Default,,0000,0000,0000,,the ratio between the area of\Nthis entire circle and the
Dialogue: 0,0:01:44.18,0:01:49.24,Default,,0000,0000,0000,,ratio of this cube right\Nhere-- or this square.
Dialogue: 0,0:01:49.24,0:01:50.33,Default,,0000,0000,0000,,I shouldn't say a cube.
Dialogue: 0,0:01:50.33,0:01:54.64,Default,,0000,0000,0000,,Cube would be if we went into\N3D --but the ratio of the area
Dialogue: 0,0:01:54.64,0:01:59.26,Default,,0000,0000,0000,,of the circle to this square\Nright here is also
Dialogue: 0,0:01:59.26,0:02:00.23,Default,,0000,0000,0000,,equal to pie.
Dialogue: 0,0:02:00.23,0:02:01.94,Default,,0000,0000,0000,,That could be actually\Nan alternate way of
Dialogue: 0,0:02:01.94,0:02:03.73,Default,,0000,0000,0000,,defining what pi is.
Dialogue: 0,0:02:03.73,0:02:06.79,Default,,0000,0000,0000,,And if you were to measure it\Nvery carefully using-- there's
Dialogue: 0,0:02:06.79,0:02:10.23,Default,,0000,0000,0000,,thousands of methods you could\Ndo it --you would get 3.14159
Dialogue: 0,0:02:10.23,0:02:12.55,Default,,0000,0000,0000,,and keep going on\Nand on and on.
Dialogue: 0,0:02:12.55,0:02:14.34,Default,,0000,0000,0000,,But we're not going to delve\Ntoo deeply into that.
Dialogue: 0,0:02:14.34,0:02:17.10,Default,,0000,0000,0000,,Maybe one day I'll make a\Nwhole play list on pi.
Dialogue: 0,0:02:17.10,0:02:19.01,Default,,0000,0000,0000,,But we just need to know that\Nthe area is equal to pi times
Dialogue: 0,0:02:19.01,0:02:21.56,Default,,0000,0000,0000,,r squared, so let's just\Napply the numbers here.
Dialogue: 0,0:02:21.56,0:02:27.60,Default,,0000,0000,0000,,So in our example, the area is\Nequal to pi times 3 meters
Dialogue: 0,0:02:27.60,0:02:34.36,Default,,0000,0000,0000,,squared, which is equal to pi\Ntimes 9 meters squared, or the
Dialogue: 0,0:02:34.36,0:02:38.10,Default,,0000,0000,0000,,conventional way to write this\Nis, equal to 9pi
Dialogue: 0,0:02:38.10,0:02:39.77,Default,,0000,0000,0000,,meters squared.
Dialogue: 0,0:02:39.77,0:02:43.21,Default,,0000,0000,0000,,And remembered 9pi, the\Nconvention is just to leave it
Dialogue: 0,0:02:43.21,0:02:48.64,Default,,0000,0000,0000,,that way, but this is the same\Nthing as 9 times 3.14159, which
Dialogue: 0,0:02:48.64,0:02:52.14,Default,,0000,0000,0000,,is probably going to be, like,\N28 point something
Dialogue: 0,0:02:52.14,0:02:52.95,Default,,0000,0000,0000,,meters squared.
Dialogue: 0,0:02:52.95,0:02:56.41,Default,,0000,0000,0000,,Just remember, this is just\Nsome number and it's not 9.
Dialogue: 0,0:02:56.41,0:02:59.78,Default,,0000,0000,0000,,It's actually closer to 28\Nbecause it's going to be
Dialogue: 0,0:02:59.78,0:03:04.05,Default,,0000,0000,0000,,9 times 3.14159, but we\Njust leave it like that.
Dialogue: 0,0:03:04.05,0:03:06.31,Default,,0000,0000,0000,,And that normally will be\Ngood enough for you to
Dialogue: 0,0:03:06.31,0:03:08.66,Default,,0000,0000,0000,,say, hey that is my area.
Dialogue: 0,0:03:08.66,0:03:10.83,Default,,0000,0000,0000,,That's my area: 9pi.
Dialogue: 0,0:03:10.83,0:03:15.51,Default,,0000,0000,0000,,Now let's go the other way:\Nlet's say I have a circle and
Dialogue: 0,0:03:15.51,0:03:20.56,Default,,0000,0000,0000,,let's say that someone would\Nsay that the area
Dialogue: 0,0:03:20.56,0:03:24.08,Default,,0000,0000,0000,,is equal to 16pi.
Dialogue: 0,0:03:24.08,0:03:32.77,Default,,0000,0000,0000,,What is the diameter of\Nthat circle going to be?
Dialogue: 0,0:03:32.77,0:03:34.84,Default,,0000,0000,0000,,Well, we know that area\Nis equal to pi times
Dialogue: 0,0:03:34.84,0:03:35.51,Default,,0000,0000,0000,,the radius squared.
Dialogue: 0,0:03:35.51,0:03:37.98,Default,,0000,0000,0000,,So at least let's\Nfigure out the radius.
Dialogue: 0,0:03:37.98,0:03:44.75,Default,,0000,0000,0000,,So the area, 16pi, is equal to\Npi times our radius squared.
Dialogue: 0,0:03:44.75,0:03:47.07,Default,,0000,0000,0000,,I'm just applying this formula.
Dialogue: 0,0:03:47.07,0:03:48.79,Default,,0000,0000,0000,,We're just going to keep\Napplying this formula
Dialogue: 0,0:03:48.79,0:03:50.54,Default,,0000,0000,0000,,over and over again when\Nwe're dealing with area.
Dialogue: 0,0:03:50.54,0:03:54.76,Default,,0000,0000,0000,,So area, which we've been\Ntold is 16pi, is equal to
Dialogue: 0,0:03:54.76,0:03:57.40,Default,,0000,0000,0000,,pi times radius squared.
Dialogue: 0,0:03:57.40,0:04:01.34,Default,,0000,0000,0000,,Now, if we divide both sides of\Nthis equation by pi, we get
Dialogue: 0,0:04:01.34,0:04:04.39,Default,,0000,0000,0000,,16 is equal to r squared.
Dialogue: 0,0:04:04.39,0:04:06.11,Default,,0000,0000,0000,,And then you take the square\Nroot of both sides and
Dialogue: 0,0:04:06.11,0:04:08.41,Default,,0000,0000,0000,,you get 4 is equal to r.
Dialogue: 0,0:04:08.41,0:04:10.86,Default,,0000,0000,0000,,I guess r could also be equal\Nto negative 4, but we're
Dialogue: 0,0:04:10.86,0:04:12.78,Default,,0000,0000,0000,,dealing with distances here;\Nyou can't have a
Dialogue: 0,0:04:12.78,0:04:14.09,Default,,0000,0000,0000,,negative radius.
Dialogue: 0,0:04:14.09,0:04:17.00,Default,,0000,0000,0000,,Or at least in the world\Nwe're living in right now.
Dialogue: 0,0:04:17.00,0:04:18.83,Default,,0000,0000,0000,,Just keep things simple;\Nwe just want to keep our
Dialogue: 0,0:04:18.83,0:04:19.90,Default,,0000,0000,0000,,distances positives.
Dialogue: 0,0:04:19.90,0:04:23.82,Default,,0000,0000,0000,,So let's say that this\Nhas a radius of 4.
Dialogue: 0,0:04:23.82,0:04:26.55,Default,,0000,0000,0000,,Now if the radius is 4,\Nwhat is its diameter?
Dialogue: 0,0:04:26.55,0:04:29.14,Default,,0000,0000,0000,,Well, the diameter is always\Ngoing to be 2 times the radius.
Dialogue: 0,0:04:29.14,0:04:31.48,Default,,0000,0000,0000,,So this 4, we're going to\Nhave another 4 there.
Dialogue: 0,0:04:31.48,0:04:35.69,Default,,0000,0000,0000,,The diameter is equal to 8.
Dialogue: 0,0:04:35.69,0:04:38.97,Default,,0000,0000,0000,,Now let's do a slightly harder\None that will kind of compound
Dialogue: 0,0:04:38.97,0:04:42.06,Default,,0000,0000,0000,,some of the other things that\Nwe've learned in the past.
Dialogue: 0,0:04:42.06,0:04:45.40,Default,,0000,0000,0000,,So let's say that I\Nhave a circle here.
Dialogue: 0,0:04:45.40,0:04:59.65,Default,,0000,0000,0000,,Let's say that its\Ncircumference is equal to 20pi,
Dialogue: 0,0:04:59.65,0:05:01.43,Default,,0000,0000,0000,,and I want to know its area.
Dialogue: 0,0:05:01.43,0:05:04.59,Default,,0000,0000,0000,,
Dialogue: 0,0:05:04.59,0:05:06.54,Default,,0000,0000,0000,,So way you do all these\Nproblems is just figure out
Dialogue: 0,0:05:06.54,0:05:09.02,Default,,0000,0000,0000,,everything you can, given what\Nthey've given you, and then
Dialogue: 0,0:05:09.02,0:05:12.01,Default,,0000,0000,0000,,maybe you can work out the\Nthing they're asking for.
Dialogue: 0,0:05:12.01,0:05:14.38,Default,,0000,0000,0000,,So if I know that the\Ncircumference is 25, what do
Dialogue: 0,0:05:14.38,0:05:16.44,Default,,0000,0000,0000,,I know about its radius?
Dialogue: 0,0:05:16.44,0:05:19.28,Default,,0000,0000,0000,,Well, we saw in the last video\Nthat the circumference is equal
Dialogue: 0,0:05:19.28,0:05:23.40,Default,,0000,0000,0000,,to 2pi times the radius.
Dialogue: 0,0:05:23.40,0:05:25.65,Default,,0000,0000,0000,,So if the circumference is\Nequal to 20pi, we could write
Dialogue: 0,0:05:25.65,0:05:30.53,Default,,0000,0000,0000,,that 20pi is the circumference\Nis equal to 2pi
Dialogue: 0,0:05:30.53,0:05:32.27,Default,,0000,0000,0000,,times the radius.
Dialogue: 0,0:05:32.27,0:05:36.60,Default,,0000,0000,0000,,Now if you divide both sides of\Nthis by pi, those cancel out.
Dialogue: 0,0:05:36.60,0:05:39.83,Default,,0000,0000,0000,,Then if you divide both sides\Nby 2, this becomes a 1, this
Dialogue: 0,0:05:39.83,0:05:43.43,Default,,0000,0000,0000,,becomes a 10, or you get\Nthe radius is equal to 10.
Dialogue: 0,0:05:43.43,0:05:44.60,Default,,0000,0000,0000,,Which makes sense, right?
Dialogue: 0,0:05:44.60,0:05:48.56,Default,,0000,0000,0000,,2pi times 10 is going\Nto be equal to 20pi.
Dialogue: 0,0:05:48.56,0:05:50.22,Default,,0000,0000,0000,,So we've figured\Nout our radius.
Dialogue: 0,0:05:50.22,0:05:57.28,Default,,0000,0000,0000,,Now, we know that the area is\Nequal to pi times r squared.
Dialogue: 0,0:05:57.28,0:05:59.86,Default,,0000,0000,0000,,And lucky for us, using the\Ncircumference, we were able
Dialogue: 0,0:05:59.86,0:06:01.42,Default,,0000,0000,0000,,to figure out the radius.
Dialogue: 0,0:06:01.42,0:06:04.69,Default,,0000,0000,0000,,Now using the radius, we\Ncan figure out the area.
Dialogue: 0,0:06:04.69,0:06:11.50,Default,,0000,0000,0000,,So the area is going to be\Nequal to pi times r squared,--
Dialogue: 0,0:06:11.50,0:06:17.04,Default,,0000,0000,0000,,r is 10 --times 10 squared,\Nwhich is equal to pi times 100.
Dialogue: 0,0:06:17.04,0:06:21.00,Default,,0000,0000,0000,,Or it's equal to 100pi.
Dialogue: 0,0:06:21.00,0:06:21.78,Default,,0000,0000,0000,,Just like that.
Dialogue: 0,0:06:21.78,0:06:25.35,Default,,0000,0000,0000,,so your circumference was 20pi,\Nwhen you went around the
Dialogue: 0,0:06:25.35,0:06:29.91,Default,,0000,0000,0000,,circle, but your area of\Nyour circle is 100pi.
Dialogue: 0,0:06:29.91,0:06:34.09,Default,,0000,0000,0000,,And if I gave you units it\Nwould be 100pi units squared.
Dialogue: 0,0:06:34.09,0:06:37.64,Default,,0000,0000,0000,,That is your area\Nright there: 100pi.
Dialogue: 0,0:06:37.64,0:06:41.17,Default,,0000,0000,0000,,Anyway, I think that's pretty\Ngood initial exposure to
Dialogue: 0,0:06:41.17,0:06:42.36,Default,,0000,0000,0000,,the area of a circle.
Dialogue: 0,0:06:42.36,0:06:44.45,Default,,0000,0000,0000,,I'll see in the next video.