[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.48,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.48,0:00:03.14,Default,,0000,0000,0000,,What I want to do in this video\Nis use some of the results from
Dialogue: 0,0:00:03.14,0:00:05.96,Default,,0000,0000,0000,,the last several videos to\Ndo some pretty neat things.
Dialogue: 0,0:00:05.96,0:00:10.00,Default,,0000,0000,0000,,So let's say this is a circle,\Nand I have an inscribed
Dialogue: 0,0:00:10.00,0:00:12.13,Default,,0000,0000,0000,,equilateral triangle\Nin this circle.
Dialogue: 0,0:00:12.13,0:00:17.34,Default,,0000,0000,0000,,So all the vertices of\Nthis triangle sit on the
Dialogue: 0,0:00:17.34,0:00:18.82,Default,,0000,0000,0000,,circumference of the circle.
Dialogue: 0,0:00:18.82,0:00:24.17,Default,,0000,0000,0000,,So I'm going to try my best to\Ndraw an equilateral triangle.
Dialogue: 0,0:00:24.17,0:00:26.96,Default,,0000,0000,0000,,I think that's about as good as\NI'm going to be able to do.
Dialogue: 0,0:00:26.96,0:00:28.62,Default,,0000,0000,0000,,And when I say equilateral\Nthat means all of these
Dialogue: 0,0:00:28.62,0:00:29.91,Default,,0000,0000,0000,,sides are the same length.
Dialogue: 0,0:00:29.91,0:00:33.06,Default,,0000,0000,0000,,So if this is side length a,\Nthen this is side length a,
Dialogue: 0,0:00:33.06,0:00:36.61,Default,,0000,0000,0000,,and that is also a\Nside of length a.
Dialogue: 0,0:00:36.61,0:00:44.01,Default,,0000,0000,0000,,And let's say we know that the\Nradius of this circle is 2.
Dialogue: 0,0:00:44.01,0:00:45.92,Default,,0000,0000,0000,,I'm just picking a number,\Njust to do this problem.
Dialogue: 0,0:00:45.92,0:00:49.60,Default,,0000,0000,0000,,So let's say the radius\Nof this circle is 2.
Dialogue: 0,0:00:49.60,0:00:51.70,Default,,0000,0000,0000,,So from the center to the\Ncircumference at any
Dialogue: 0,0:00:51.70,0:00:55.91,Default,,0000,0000,0000,,point, this distance, the\Nradius, is equal to 2.
Dialogue: 0,0:00:55.91,0:01:01.78,Default,,0000,0000,0000,,Now, what I'm going to ask you\Nis using some of the results of
Dialogue: 0,0:01:01.78,0:01:04.02,Default,,0000,0000,0000,,the last few videos and a\Nlittle bit of basic
Dialogue: 0,0:01:04.02,0:01:06.94,Default,,0000,0000,0000,,trigonometry-- and if the word\N"trigonometry" scares you,
Dialogue: 0,0:01:06.94,0:01:09.57,Default,,0000,0000,0000,,you'll just need to know maybe\Nthe first two or three videos
Dialogue: 0,0:01:09.57,0:01:11.71,Default,,0000,0000,0000,,in the trigonometry playlist to\Nbe able to understand
Dialogue: 0,0:01:11.71,0:01:12.84,Default,,0000,0000,0000,,what I do here.
Dialogue: 0,0:01:12.84,0:01:18.83,Default,,0000,0000,0000,,What I want to do is figure out\Nthe area of the region inside
Dialogue: 0,0:01:18.83,0:01:21.08,Default,,0000,0000,0000,,the circle and outside\Nof the triangle.
Dialogue: 0,0:01:21.08,0:01:25.69,Default,,0000,0000,0000,,So I want to figure out the\Narea of that little space, that
Dialogue: 0,0:01:25.69,0:01:30.94,Default,,0000,0000,0000,,space, and this space combined.
Dialogue: 0,0:01:30.94,0:01:33.49,Default,,0000,0000,0000,,So the obvious way to do\Nthis is to say, well I can
Dialogue: 0,0:01:33.49,0:01:36.67,Default,,0000,0000,0000,,figure out the area of\Nthe circle pretty easily.
Dialogue: 0,0:01:36.67,0:01:40.22,Default,,0000,0000,0000,,Area of the circle.
Dialogue: 0,0:01:40.22,0:01:43.74,Default,,0000,0000,0000,,And that's going to be\Nequal to pi r squared.
Dialogue: 0,0:01:43.74,0:01:48.84,Default,,0000,0000,0000,,Or pi times 2 squared,\Nwhich is equal to 4 pi.
Dialogue: 0,0:01:48.84,0:01:53.04,Default,,0000,0000,0000,,And I could subtract from 4\Npi the area of the triangle.
Dialogue: 0,0:01:53.04,0:01:55.45,Default,,0000,0000,0000,,So we need to figure out\Nthe area of the triangle.
Dialogue: 0,0:01:55.45,0:02:00.76,Default,,0000,0000,0000,,What is the area\Nof the triangle?
Dialogue: 0,0:02:00.76,0:02:03.93,Default,,0000,0000,0000,,
Dialogue: 0,0:02:03.93,0:02:07.26,Default,,0000,0000,0000,,Well, from several videos ago I\Nshowed you Heron's formula,
Dialogue: 0,0:02:07.26,0:02:10.72,Default,,0000,0000,0000,,where if you know the lengths\Nof the sides of a triangle
Dialogue: 0,0:02:10.72,0:02:12.07,Default,,0000,0000,0000,,you can figure out the area.
Dialogue: 0,0:02:12.07,0:02:14.18,Default,,0000,0000,0000,,But we don't know the lengths\Nof the sides just yet.
Dialogue: 0,0:02:14.18,0:02:16.56,Default,,0000,0000,0000,,Once we do maybe we can\Nfigure out the area.
Dialogue: 0,0:02:16.56,0:02:18.74,Default,,0000,0000,0000,,Let me apply Heron's\Nformula not knowing it.
Dialogue: 0,0:02:18.74,0:02:21.95,Default,,0000,0000,0000,,So let me just say that the\Nlengths of this equilateral--
Dialogue: 0,0:02:21.95,0:02:23.76,Default,,0000,0000,0000,,the lengths of the\Nsides-- are a.
Dialogue: 0,0:02:23.76,0:02:31.45,Default,,0000,0000,0000,,Applying Heron's formula, we\Nfirst define our variable
Dialogue: 0,0:02:31.45,0:02:38.22,Default,,0000,0000,0000,,s as being equal to a\Nplus a plus a, over 2.
Dialogue: 0,0:02:38.22,0:02:42.07,Default,,0000,0000,0000,,Or that's the same\Nthing as 3a over 2.
Dialogue: 0,0:02:42.07,0:02:46.38,Default,,0000,0000,0000,,And then the area of this\Ntriangle, in terms of a.
Dialogue: 0,0:02:46.38,0:02:52.91,Default,,0000,0000,0000,,So the area is going to be\Nequal to the square root of
Dialogue: 0,0:02:52.91,0:02:59.31,Default,,0000,0000,0000,,s, which is 3a over\N2, times s minus a.
Dialogue: 0,0:02:59.31,0:03:03.82,Default,,0000,0000,0000,,So that's 3a over 2 minus a.
Dialogue: 0,0:03:03.82,0:03:07.06,Default,,0000,0000,0000,,Or I could just\Nwrite, 2a over 2.
Dialogue: 0,0:03:07.06,0:03:08.97,Default,,0000,0000,0000,,Right? a is the same\Nthing as 2a over 2.
Dialogue: 0,0:03:08.97,0:03:10.74,Default,,0000,0000,0000,,You could cancel\Nthose out and get a.
Dialogue: 0,0:03:10.74,0:03:13.17,Default,,0000,0000,0000,,And then I'm going to\Ndo that three times.
Dialogue: 0,0:03:13.17,0:03:16.00,Default,,0000,0000,0000,,So instead of just multiplying\Nthat out three times for each
Dialogue: 0,0:03:16.00,0:03:18.64,Default,,0000,0000,0000,,of the sides, by Heron's\Nformula I could just say
Dialogue: 0,0:03:18.64,0:03:20.70,Default,,0000,0000,0000,,to the third power.
Dialogue: 0,0:03:20.70,0:03:22.00,Default,,0000,0000,0000,,So what's this going\Nto be equal to?
Dialogue: 0,0:03:22.00,0:03:31.05,Default,,0000,0000,0000,,This is going to be equal to\Nthe square root of 3a over 2.
Dialogue: 0,0:03:31.05,0:03:34.07,Default,,0000,0000,0000,,And then this right here\Nis going to be equal
Dialogue: 0,0:03:34.07,0:03:36.81,Default,,0000,0000,0000,,to 3a minus 2a, is a.
Dialogue: 0,0:03:36.81,0:03:42.01,Default,,0000,0000,0000,,So a/2 to the third power.
Dialogue: 0,0:03:42.01,0:03:44.86,Default,,0000,0000,0000,,And so this is going to be\Nequal to-- I'll arbitrarily
Dialogue: 0,0:03:44.86,0:03:46.49,Default,,0000,0000,0000,,switch colors.
Dialogue: 0,0:03:46.49,0:03:53.56,Default,,0000,0000,0000,,We have 3a times a to the\Nthird, which is 3a to the
Dialogue: 0,0:03:53.56,0:03:58.17,Default,,0000,0000,0000,,fourth, over 2 times\N2 to the third.
Dialogue: 0,0:03:58.17,0:04:03.40,Default,,0000,0000,0000,,Well that's 2 to the\Nfourth power, or 16.
Dialogue: 0,0:04:03.40,0:04:03.68,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:04:03.68,0:04:07.10,Default,,0000,0000,0000,,2 times 2 to the third\Nis 2 to the fourth.
Dialogue: 0,0:04:07.10,0:04:07.89,Default,,0000,0000,0000,,That's 16.
Dialogue: 0,0:04:07.89,0:04:10.66,Default,,0000,0000,0000,,And then if we take the square\Nroot of the numerator and the
Dialogue: 0,0:04:10.66,0:04:14.15,Default,,0000,0000,0000,,denominator, this is going to\Nbe equal to the square root of
Dialogue: 0,0:04:14.15,0:04:16.69,Default,,0000,0000,0000,,a to the fourth is a squared.
Dialogue: 0,0:04:16.69,0:04:21.39,Default,,0000,0000,0000,,a squared times, well I'll just\Nwrite the square root of 3,
Dialogue: 0,0:04:21.39,0:04:24.86,Default,,0000,0000,0000,,over the square root of the\Ndenominator, which is just 4.
Dialogue: 0,0:04:24.86,0:04:30.13,Default,,0000,0000,0000,,So if we know a, using Heron's\Nformula we know what the area
Dialogue: 0,0:04:30.13,0:04:32.72,Default,,0000,0000,0000,,of this equilateral\Ntriangle is.
Dialogue: 0,0:04:32.72,0:04:35.08,Default,,0000,0000,0000,,So how can we figure out a?
Dialogue: 0,0:04:35.08,0:04:37.77,Default,,0000,0000,0000,,So what else do we know about\Nequilateral triangles?
Dialogue: 0,0:04:37.77,0:04:42.69,Default,,0000,0000,0000,,Well we know that all of\Nthese angles are equal.
Dialogue: 0,0:04:42.69,0:04:45.72,Default,,0000,0000,0000,,And since they must add\Nup to 180 degrees, they
Dialogue: 0,0:04:45.72,0:04:48.21,Default,,0000,0000,0000,,all must be 60 degrees.
Dialogue: 0,0:04:48.21,0:04:51.89,Default,,0000,0000,0000,,That's 60 degrees,\Nthat's 60 degrees, and
Dialogue: 0,0:04:51.89,0:04:54.09,Default,,0000,0000,0000,,that is 60 degrees.
Dialogue: 0,0:04:54.09,0:04:56.98,Default,,0000,0000,0000,,Now let's see if we can use the\Nlast video, where I talked
Dialogue: 0,0:04:56.98,0:05:01.72,Default,,0000,0000,0000,,about the relationship\Nbetween an inscribed angle
Dialogue: 0,0:05:01.72,0:05:02.80,Default,,0000,0000,0000,,and a central angle.
Dialogue: 0,0:05:02.80,0:05:04.56,Default,,0000,0000,0000,,So this is an inscribed\Nangle right here.
Dialogue: 0,0:05:04.56,0:05:09.62,Default,,0000,0000,0000,,It's vertex is sitting\Non the circumference.
Dialogue: 0,0:05:09.62,0:05:16.63,Default,,0000,0000,0000,,And so it is subtending\Nthis arc right here.
Dialogue: 0,0:05:16.63,0:05:20.50,Default,,0000,0000,0000,,
Dialogue: 0,0:05:20.50,0:05:25.00,Default,,0000,0000,0000,,And the central angle that\Nis subtending that same arc
Dialogue: 0,0:05:25.00,0:05:26.33,Default,,0000,0000,0000,,is this one right here.
Dialogue: 0,0:05:26.33,0:05:29.88,Default,,0000,0000,0000,,
Dialogue: 0,0:05:29.88,0:05:33.74,Default,,0000,0000,0000,,The central angles subtending\Nthat same arc is that
Dialogue: 0,0:05:33.74,0:05:34.96,Default,,0000,0000,0000,,one right there.
Dialogue: 0,0:05:34.96,0:05:39.17,Default,,0000,0000,0000,,So based on what we saw in the\Nlast video, the central angle
Dialogue: 0,0:05:39.17,0:05:41.98,Default,,0000,0000,0000,,that subtends the same arc is\Ngoing to be double of
Dialogue: 0,0:05:41.98,0:05:43.04,Default,,0000,0000,0000,,the inscribed angle.
Dialogue: 0,0:05:43.04,0:05:47.23,Default,,0000,0000,0000,,So this angle right here is\Ngoing to be 120 degrees.
Dialogue: 0,0:05:47.23,0:05:48.86,Default,,0000,0000,0000,,Let me just put an arrow there.
Dialogue: 0,0:05:48.86,0:05:50.86,Default,,0000,0000,0000,,120 degrees.
Dialogue: 0,0:05:50.86,0:05:52.44,Default,,0000,0000,0000,,It's double of that one.
Dialogue: 0,0:05:52.44,0:05:56.11,Default,,0000,0000,0000,,Now, if I were to exactly\Nbisect this angle right here.
Dialogue: 0,0:05:56.11,0:05:58.14,Default,,0000,0000,0000,,So I go halfway through the\Nangle, and I want to just go
Dialogue: 0,0:05:58.14,0:06:01.26,Default,,0000,0000,0000,,straight down like that.
Dialogue: 0,0:06:01.26,0:06:03.31,Default,,0000,0000,0000,,What are these two\Nangles going to be?
Dialogue: 0,0:06:03.31,0:06:04.44,Default,,0000,0000,0000,,Well, they're going\Nto be 60 degrees.
Dialogue: 0,0:06:04.44,0:06:05.76,Default,,0000,0000,0000,,I'm bisecting that angle.
Dialogue: 0,0:06:05.76,0:06:10.48,Default,,0000,0000,0000,,That is 60 degrees, and that\Nis 60 degrees right there.
Dialogue: 0,0:06:10.48,0:06:14.45,Default,,0000,0000,0000,,And we know that I'm\Nsplitting this side in two.
Dialogue: 0,0:06:14.45,0:06:17.08,Default,,0000,0000,0000,,This is an isosceles triangle.
Dialogue: 0,0:06:17.08,0:06:19.04,Default,,0000,0000,0000,,This is a radius right here.
Dialogue: 0,0:06:19.04,0:06:21.03,Default,,0000,0000,0000,,Radius r is equal to 2.
Dialogue: 0,0:06:21.03,0:06:24.53,Default,,0000,0000,0000,,This is a radius right\Nhere of r is equal to 2.
Dialogue: 0,0:06:24.53,0:06:26.09,Default,,0000,0000,0000,,So this whole triangle\Nis symmetric.
Dialogue: 0,0:06:26.09,0:06:28.54,Default,,0000,0000,0000,,If I go straight down the\Nmiddle, this length right
Dialogue: 0,0:06:28.54,0:06:33.10,Default,,0000,0000,0000,,here is going to be\Nthat side divided by 2.
Dialogue: 0,0:06:33.10,0:06:36.28,Default,,0000,0000,0000,,That side right there is going\Nto be that side divided by 2.
Dialogue: 0,0:06:36.28,0:06:37.24,Default,,0000,0000,0000,,Let me draw that over here.
Dialogue: 0,0:06:37.24,0:06:39.89,Default,,0000,0000,0000,,If I just take an isosceles\Ntriangle, any isosceles
Dialogue: 0,0:06:39.89,0:06:44.85,Default,,0000,0000,0000,,triangle, where this side is\Nequivalent to that side.
Dialogue: 0,0:06:44.85,0:06:47.30,Default,,0000,0000,0000,,Those are our radiuses\Nin this example.
Dialogue: 0,0:06:47.30,0:06:49.53,Default,,0000,0000,0000,,And this angle is going to\Nbe equal to that angle.
Dialogue: 0,0:06:49.53,0:06:51.79,Default,,0000,0000,0000,,If I were to just go straight\Ndown this angle right
Dialogue: 0,0:06:51.79,0:06:55.26,Default,,0000,0000,0000,,here, I would split that\Nopposite side in two.
Dialogue: 0,0:06:55.26,0:06:56.88,Default,,0000,0000,0000,,So these two lengths\Nare going to be equal.
Dialogue: 0,0:06:56.88,0:06:59.12,Default,,0000,0000,0000,,In this case if the whole\Nthing is a, each of these
Dialogue: 0,0:06:59.12,0:07:01.14,Default,,0000,0000,0000,,are going to be a/2.
Dialogue: 0,0:07:01.14,0:07:04.42,Default,,0000,0000,0000,,Now, let's see if we can use\Nthis and a little bit of
Dialogue: 0,0:07:04.42,0:07:08.62,Default,,0000,0000,0000,,trigonometry to find the\Nrelationship between a and r.
Dialogue: 0,0:07:08.62,0:07:12.05,Default,,0000,0000,0000,,Because if we're able to solve\Nfor a using r, then we can then
Dialogue: 0,0:07:12.05,0:07:14.64,Default,,0000,0000,0000,,put that value of a in here and\Nwe'll get the area
Dialogue: 0,0:07:14.64,0:07:15.69,Default,,0000,0000,0000,,of our triangle.
Dialogue: 0,0:07:15.69,0:07:17.60,Default,,0000,0000,0000,,And then we could subtract\Nthat from the area of the
Dialogue: 0,0:07:17.60,0:07:20.07,Default,,0000,0000,0000,,circle, and we're done.
Dialogue: 0,0:07:20.07,0:07:22.05,Default,,0000,0000,0000,,We will have solved\Nthe problem.
Dialogue: 0,0:07:22.05,0:07:24.61,Default,,0000,0000,0000,,So let's see if we can do that.
Dialogue: 0,0:07:24.61,0:07:29.34,Default,,0000,0000,0000,,So we have an angle\Nhere of 60 degrees.
Dialogue: 0,0:07:29.34,0:07:32.05,Default,,0000,0000,0000,,Half of this whole central\Nangle right there.
Dialogue: 0,0:07:32.05,0:07:35.89,Default,,0000,0000,0000,,If this angle is 60 degrees,\Nwe have a/2 that's
Dialogue: 0,0:07:35.89,0:07:37.38,Default,,0000,0000,0000,,opposite to this angle.
Dialogue: 0,0:07:37.38,0:07:43.48,Default,,0000,0000,0000,,So we have an opposite\Nis equal to a/2.
Dialogue: 0,0:07:43.48,0:07:44.86,Default,,0000,0000,0000,,And we also have\Nthe hypotenuse.
Dialogue: 0,0:07:44.86,0:07:45.04,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:07:45.04,0:07:46.87,Default,,0000,0000,0000,,This is a right\Ntriangle right here.
Dialogue: 0,0:07:46.87,0:07:49.83,Default,,0000,0000,0000,,You're just going straight\Ndown, and you're bisecting
Dialogue: 0,0:07:49.83,0:07:50.84,Default,,0000,0000,0000,,that opposite side.
Dialogue: 0,0:07:50.84,0:07:52.64,Default,,0000,0000,0000,,This is a right triangle.
Dialogue: 0,0:07:52.64,0:07:54.00,Default,,0000,0000,0000,,So we can do a little\Ntrigonometry.
Dialogue: 0,0:07:54.00,0:08:02.55,Default,,0000,0000,0000,,Our opposite is a/2, the\Nhypotenuse is equal to r.
Dialogue: 0,0:08:02.55,0:08:05.02,Default,,0000,0000,0000,,This is the hypotenuse, right\Nhere, of our right triangle.
Dialogue: 0,0:08:05.02,0:08:06.36,Default,,0000,0000,0000,,So that is equal to 2.
Dialogue: 0,0:08:06.36,0:08:12.44,Default,,0000,0000,0000,,So what trig ratio is the\Nratio of an angle's opposite
Dialogue: 0,0:08:12.44,0:08:14.92,Default,,0000,0000,0000,,side to hypotenuse?
Dialogue: 0,0:08:14.92,0:08:18.91,Default,,0000,0000,0000,,So some of you all might get\Ntired of me doing this all
Dialogue: 0,0:08:18.91,0:08:22.03,Default,,0000,0000,0000,,the time, but SOH CAH TOA.
Dialogue: 0,0:08:22.03,0:08:27.07,Default,,0000,0000,0000,,SOH-- sin of an angle is\Nequal to the opposite
Dialogue: 0,0:08:27.07,0:08:28.62,Default,,0000,0000,0000,,over the hypotenuse.
Dialogue: 0,0:08:28.62,0:08:29.58,Default,,0000,0000,0000,,So let me scroll\Ndown a little bit.
Dialogue: 0,0:08:29.58,0:08:31.27,Default,,0000,0000,0000,,I'm running out of space.
Dialogue: 0,0:08:31.27,0:08:38.70,Default,,0000,0000,0000,,So the sin of this angle right\Nhere, the sin of 60 degrees, is
Dialogue: 0,0:08:38.70,0:08:42.07,Default,,0000,0000,0000,,going to be equal to the\Nopposite side, is going to be
Dialogue: 0,0:08:42.07,0:08:45.80,Default,,0000,0000,0000,,equal to a/2, over the\Nhypotenuse, which is
Dialogue: 0,0:08:45.80,0:08:48.14,Default,,0000,0000,0000,,our radius-- over 2.
Dialogue: 0,0:08:48.14,0:08:54.51,Default,,0000,0000,0000,,Which is equal to a/2\Ndivided by 2 is a/4.
Dialogue: 0,0:08:54.51,0:08:56.88,Default,,0000,0000,0000,,And what is sin of 60 degrees?
Dialogue: 0,0:08:56.88,0:08:59.72,Default,,0000,0000,0000,,And if the word "sin" looks\Ncompletely foreign to you,
Dialogue: 0,0:08:59.72,0:09:04.15,Default,,0000,0000,0000,,watch the first several videos\Non the trigonometry playlist.
Dialogue: 0,0:09:04.15,0:09:06.24,Default,,0000,0000,0000,,It shouldn't be too daunting.
Dialogue: 0,0:09:06.24,0:09:08.31,Default,,0000,0000,0000,,sin of 60 degrees you\Nmight remember from your
Dialogue: 0,0:09:08.31,0:09:10.68,Default,,0000,0000,0000,,30-60-90 triangles.
Dialogue: 0,0:09:10.68,0:09:13.21,Default,,0000,0000,0000,,So let me draw one right there.
Dialogue: 0,0:09:13.21,0:09:15.70,Default,,0000,0000,0000,,So that is a 30-60-90 triangle.
Dialogue: 0,0:09:15.70,0:09:21.54,Default,,0000,0000,0000,,If this is 60 degrees, that\Nis 30 degrees, that is 90.
Dialogue: 0,0:09:21.54,0:09:26.66,Default,,0000,0000,0000,,You might remember that this is\Nof length 1, this is going to
Dialogue: 0,0:09:26.66,0:09:29.52,Default,,0000,0000,0000,,be of length 1/2, and this\Nis going to be of length
Dialogue: 0,0:09:29.52,0:09:31.37,Default,,0000,0000,0000,,square root of 3 over 2.
Dialogue: 0,0:09:31.37,0:09:35.30,Default,,0000,0000,0000,,So the sin of 60 degrees is\Nopposite over hypotenuse.
Dialogue: 0,0:09:35.30,0:09:37.77,Default,,0000,0000,0000,,Square root of 3 over 2 over 1.
Dialogue: 0,0:09:37.77,0:09:40.94,Default,,0000,0000,0000,,sin of 60 degrees.
Dialogue: 0,0:09:40.94,0:09:42.84,Default,,0000,0000,0000,,If you don't have a calculator,\Nyou could just use this--
Dialogue: 0,0:09:42.84,0:09:44.93,Default,,0000,0000,0000,,is square root of 3 over 2.
Dialogue: 0,0:09:44.93,0:09:48.89,Default,,0000,0000,0000,,So this right here is\Nsquare root of 3 over 2.
Dialogue: 0,0:09:48.89,0:09:51.28,Default,,0000,0000,0000,,Now we can solve for a.
Dialogue: 0,0:09:51.28,0:09:56.92,Default,,0000,0000,0000,,Square root of 3 over\N2 is equal to a/4.
Dialogue: 0,0:09:56.92,0:09:59.61,Default,,0000,0000,0000,,Let's multiply both sides by 4.
Dialogue: 0,0:09:59.61,0:10:01.64,Default,,0000,0000,0000,,So you get this 4 cancels out.
Dialogue: 0,0:10:01.64,0:10:03.44,Default,,0000,0000,0000,,You multiply 4 here.
Dialogue: 0,0:10:03.44,0:10:04.48,Default,,0000,0000,0000,,This becomes a 2.
Dialogue: 0,0:10:04.48,0:10:05.66,Default,,0000,0000,0000,,This becomes a 1.
Dialogue: 0,0:10:05.66,0:10:09.25,Default,,0000,0000,0000,,You get a is equal to\N2 square roots of 3.
Dialogue: 0,0:10:09.25,0:10:11.00,Default,,0000,0000,0000,,We're in the home stretch.
Dialogue: 0,0:10:11.00,0:10:15.42,Default,,0000,0000,0000,,We just figured out the length\Nof each of these sides.
Dialogue: 0,0:10:15.42,0:10:17.46,Default,,0000,0000,0000,,We used Heron's formula to\Nfigure out the area of the
Dialogue: 0,0:10:17.46,0:10:19.08,Default,,0000,0000,0000,,triangle in terms\Nof those lengths.
Dialogue: 0,0:10:19.08,0:10:22.11,Default,,0000,0000,0000,,So we just substitute this\Nvalue of a into there
Dialogue: 0,0:10:22.11,0:10:24.67,Default,,0000,0000,0000,,to get our actual area.
Dialogue: 0,0:10:24.67,0:10:30.38,Default,,0000,0000,0000,,So our triangle's area\Nis equal to a squared.
Dialogue: 0,0:10:30.38,0:10:31.69,Default,,0000,0000,0000,,What's a squared?
Dialogue: 0,0:10:31.69,0:10:37.78,Default,,0000,0000,0000,,That is 2 square roots\Nof 3 squared, times the
Dialogue: 0,0:10:37.78,0:10:42.71,Default,,0000,0000,0000,,square root of 3 over 4.
Dialogue: 0,0:10:42.71,0:10:45.47,Default,,0000,0000,0000,,We just did a squared times\Nthe square root of 3 over 4.
Dialogue: 0,0:10:45.47,0:10:51.95,Default,,0000,0000,0000,,This is going to be equal\Nto 4 times 3 times the
Dialogue: 0,0:10:51.95,0:10:53.93,Default,,0000,0000,0000,,square of 3 over 4.
Dialogue: 0,0:10:53.93,0:10:55.35,Default,,0000,0000,0000,,These 4's cancel.
Dialogue: 0,0:10:55.35,0:10:58.36,Default,,0000,0000,0000,,So the area of our triangle\Nwe got is 3 times the
Dialogue: 0,0:10:58.36,0:11:00.77,Default,,0000,0000,0000,,square root of 3.
Dialogue: 0,0:11:00.77,0:11:03.16,Default,,0000,0000,0000,,So the area here is 3\Nsquare roots of 3.
Dialogue: 0,0:11:03.16,0:11:06.48,Default,,0000,0000,0000,,That's the area of\Nthis entire triangle.
Dialogue: 0,0:11:06.48,0:11:08.81,Default,,0000,0000,0000,,Now, to go back to what this\Nquestion was all about.
Dialogue: 0,0:11:08.81,0:11:12.97,Default,,0000,0000,0000,,The area of this orange area\Noutside of the triangle
Dialogue: 0,0:11:12.97,0:11:14.53,Default,,0000,0000,0000,,and inside of the circle.
Dialogue: 0,0:11:14.53,0:11:18.46,Default,,0000,0000,0000,,Well, the area of\Nour circle is 4 pi.
Dialogue: 0,0:11:18.46,0:11:23.34,Default,,0000,0000,0000,,And from that we subtract\Nthe area of the triangle,
Dialogue: 0,0:11:23.34,0:11:25.17,Default,,0000,0000,0000,,3 square roots of 3.
Dialogue: 0,0:11:25.17,0:11:27.39,Default,,0000,0000,0000,,And we are done.
Dialogue: 0,0:11:27.39,0:11:28.67,Default,,0000,0000,0000,,This is our answer.
Dialogue: 0,0:11:28.67,0:11:35.27,Default,,0000,0000,0000,,This is the area of this\Norange region right there.
Dialogue: 0,0:11:35.27,0:11:37.80,Default,,0000,0000,0000,,Anyway, hopefully\Nyou found that fun.
Dialogue: 0,0:11:37.80,0:11:38.04,Default,,0000,0000,0000,,