0:00:00.000,0:00:00.480


0:00:00.480,0:00:03.140
What I want to do in this video[br]is use some of the results from

0:00:03.140,0:00:05.960
the last several videos to[br]do some pretty neat things.

0:00:05.960,0:00:10.000
So let's say this is a circle,[br]and I have an inscribed

0:00:10.000,0:00:12.130
equilateral triangle[br]in this circle.

0:00:12.130,0:00:17.340
So all the vertices of[br]this triangle sit on the

0:00:17.340,0:00:18.825
circumference of the circle.

0:00:18.825,0:00:24.170
So I'm going to try my best to[br]draw an equilateral triangle.

0:00:24.170,0:00:26.960
I think that's about as good as[br]I'm going to be able to do.

0:00:26.960,0:00:28.620
And when I say equilateral[br]that means all of these

0:00:28.620,0:00:29.910
sides are the same length.

0:00:29.910,0:00:33.060
So if this is side length a,[br]then this is side length a,

0:00:33.060,0:00:36.610
and that is also a[br]side of length a.

0:00:36.610,0:00:44.010
And let's say we know that the[br]radius of this circle is 2.

0:00:44.010,0:00:45.925
I'm just picking a number,[br]just to do this problem.

0:00:45.925,0:00:49.600
So let's say the radius[br]of this circle is 2.

0:00:49.600,0:00:51.700
So from the center to the[br]circumference at any

0:00:51.700,0:00:55.910
point, this distance, the[br]radius, is equal to 2.

0:00:55.910,0:01:01.780
Now, what I'm going to ask you[br]is using some of the results of

0:01:01.780,0:01:04.020
the last few videos and a[br]little bit of basic

0:01:04.020,0:01:06.940
trigonometry-- and if the word[br]"trigonometry" scares you,

0:01:06.940,0:01:09.570
you'll just need to know maybe[br]the first two or three videos

0:01:09.570,0:01:11.710
in the trigonometry playlist to[br]be able to understand

0:01:11.710,0:01:12.840
what I do here.

0:01:12.840,0:01:18.830
What I want to do is figure out[br]the area of the region inside

0:01:18.830,0:01:21.080
the circle and outside[br]of the triangle.

0:01:21.080,0:01:25.690
So I want to figure out the[br]area of that little space, that

0:01:25.690,0:01:30.940
space, and this space combined.

0:01:30.940,0:01:33.490
So the obvious way to do[br]this is to say, well I can

0:01:33.490,0:01:36.670
figure out the area of[br]the circle pretty easily.

0:01:36.670,0:01:40.215
Area of the circle.

0:01:40.215,0:01:43.740
And that's going to be[br]equal to pi r squared.

0:01:43.740,0:01:48.840
Or pi times 2 squared,[br]which is equal to 4 pi.

0:01:48.840,0:01:53.040
And I could subtract from 4[br]pi the area of the triangle.

0:01:53.040,0:01:55.450
So we need to figure out[br]the area of the triangle.

0:01:55.450,0:02:00.760
What is the area[br]of the triangle?

0:02:00.760,0:02:03.930


0:02:03.930,0:02:07.260
Well, from several videos ago I[br]showed you Heron's formula,

0:02:07.260,0:02:10.720
where if you know the lengths[br]of the sides of a triangle

0:02:10.720,0:02:12.070
you can figure out the area.

0:02:12.070,0:02:14.180
But we don't know the lengths[br]of the sides just yet.

0:02:14.180,0:02:16.560
Once we do maybe we can[br]figure out the area.

0:02:16.560,0:02:18.740
Let me apply Heron's[br]formula not knowing it.

0:02:18.740,0:02:21.950
So let me just say that the[br]lengths of this equilateral--

0:02:21.950,0:02:23.760
the lengths of the[br]sides-- are a.

0:02:23.760,0:02:31.450
Applying Heron's formula, we[br]first define our variable

0:02:31.450,0:02:38.220
s as being equal to a[br]plus a plus a, over 2.

0:02:38.220,0:02:42.070
Or that's the same[br]thing as 3a over 2.

0:02:42.070,0:02:46.380
And then the area of this[br]triangle, in terms of a.

0:02:46.380,0:02:52.910
So the area is going to be[br]equal to the square root of

0:02:52.910,0:02:59.310
s, which is 3a over[br]2, times s minus a.

0:02:59.310,0:03:03.820
So that's 3a over 2 minus a.

0:03:03.820,0:03:07.060
Or I could just[br]write, 2a over 2.

0:03:07.060,0:03:08.970
Right? a is the same[br]thing as 2a over 2.

0:03:08.970,0:03:10.740
You could cancel[br]those out and get a.

0:03:10.740,0:03:13.170
And then I'm going to[br]do that three times.

0:03:13.170,0:03:16.000
So instead of just multiplying[br]that out three times for each

0:03:16.000,0:03:18.640
of the sides, by Heron's[br]formula I could just say

0:03:18.640,0:03:20.700
to the third power.

0:03:20.700,0:03:22.000
So what's this going[br]to be equal to?

0:03:22.000,0:03:31.050
This is going to be equal to[br]the square root of 3a over 2.

0:03:31.050,0:03:34.070
And then this right here[br]is going to be equal

0:03:34.070,0:03:36.810
to 3a minus 2a, is a.

0:03:36.810,0:03:42.010
So a/2 to the third power.

0:03:42.010,0:03:44.860
And so this is going to be[br]equal to-- I'll arbitrarily

0:03:44.860,0:03:46.490
switch colors.

0:03:46.490,0:03:53.560
We have 3a times a to the[br]third, which is 3a to the

0:03:53.560,0:03:58.170
fourth, over 2 times[br]2 to the third.

0:03:58.170,0:04:03.400
Well that's 2 to the[br]fourth power, or 16.

0:04:03.400,0:04:03.680
Right?

0:04:03.680,0:04:07.100
2 times 2 to the third[br]is 2 to the fourth.

0:04:07.100,0:04:07.890
That's 16.

0:04:07.890,0:04:10.660
And then if we take the square[br]root of the numerator and the

0:04:10.660,0:04:14.150
denominator, this is going to[br]be equal to the square root of

0:04:14.150,0:04:16.690
a to the fourth is a squared.

0:04:16.690,0:04:21.390
a squared times, well I'll just[br]write the square root of 3,

0:04:21.390,0:04:24.860
over the square root of the[br]denominator, which is just 4.

0:04:24.860,0:04:30.130
So if we know a, using Heron's[br]formula we know what the area

0:04:30.130,0:04:32.720
of this equilateral[br]triangle is.

0:04:32.720,0:04:35.080
So how can we figure out a?

0:04:35.080,0:04:37.770
So what else do we know about[br]equilateral triangles?

0:04:37.770,0:04:42.690
Well we know that all of[br]these angles are equal.

0:04:42.690,0:04:45.720
And since they must add[br]up to 180 degrees, they

0:04:45.720,0:04:48.210
all must be 60 degrees.

0:04:48.210,0:04:51.890
That's 60 degrees,[br]that's 60 degrees, and

0:04:51.890,0:04:54.090
that is 60 degrees.

0:04:54.090,0:04:56.980
Now let's see if we can use the[br]last video, where I talked

0:04:56.980,0:05:01.720
about the relationship[br]between an inscribed angle

0:05:01.720,0:05:02.800
and a central angle.

0:05:02.800,0:05:04.560
So this is an inscribed[br]angle right here.

0:05:04.560,0:05:09.620
It's vertex is sitting[br]on the circumference.

0:05:09.620,0:05:16.630
And so it is subtending[br]this arc right here.

0:05:16.630,0:05:20.500


0:05:20.500,0:05:25.000
And the central angle that[br]is subtending that same arc

0:05:25.000,0:05:26.330
is this one right here.

0:05:26.330,0:05:29.880


0:05:29.880,0:05:33.740
The central angles subtending[br]that same arc is that

0:05:33.740,0:05:34.960
one right there.

0:05:34.960,0:05:39.170
So based on what we saw in the[br]last video, the central angle

0:05:39.170,0:05:41.980
that subtends the same arc is[br]going to be double of

0:05:41.980,0:05:43.040
the inscribed angle.

0:05:43.040,0:05:47.230
So this angle right here is[br]going to be 120 degrees.

0:05:47.230,0:05:48.860
Let me just put an arrow there.

0:05:48.860,0:05:50.860
120 degrees.

0:05:50.860,0:05:52.440
It's double of that one.

0:05:52.440,0:05:56.110
Now, if I were to exactly[br]bisect this angle right here.

0:05:56.110,0:05:58.140
So I go halfway through the[br]angle, and I want to just go

0:05:58.140,0:06:01.260
straight down like that.

0:06:01.260,0:06:03.310
What are these two[br]angles going to be?

0:06:03.310,0:06:04.440
Well, they're going[br]to be 60 degrees.

0:06:04.440,0:06:05.760
I'm bisecting that angle.

0:06:05.760,0:06:10.480
That is 60 degrees, and that[br]is 60 degrees right there.

0:06:10.480,0:06:14.450
And we know that I'm[br]splitting this side in two.

0:06:14.450,0:06:17.080
This is an isosceles triangle.

0:06:17.080,0:06:19.040
This is a radius right here.

0:06:19.040,0:06:21.030
Radius r is equal to 2.

0:06:21.030,0:06:24.530
This is a radius right[br]here of r is equal to 2.

0:06:24.530,0:06:26.090
So this whole triangle[br]is symmetric.

0:06:26.090,0:06:28.540
If I go straight down the[br]middle, this length right

0:06:28.540,0:06:33.100
here is going to be[br]that side divided by 2.

0:06:33.100,0:06:36.280
That side right there is going[br]to be that side divided by 2.

0:06:36.280,0:06:37.240
Let me draw that over here.

0:06:37.240,0:06:39.890
If I just take an isosceles[br]triangle, any isosceles

0:06:39.890,0:06:44.850
triangle, where this side is[br]equivalent to that side.

0:06:44.850,0:06:47.300
Those are our radiuses[br]in this example.

0:06:47.300,0:06:49.530
And this angle is going to[br]be equal to that angle.

0:06:49.530,0:06:51.790
If I were to just go straight[br]down this angle right

0:06:51.790,0:06:55.260
here, I would split that[br]opposite side in two.

0:06:55.260,0:06:56.880
So these two lengths[br]are going to be equal.

0:06:56.880,0:06:59.120
In this case if the whole[br]thing is a, each of these

0:06:59.120,0:07:01.140
are going to be a/2.

0:07:01.140,0:07:04.420
Now, let's see if we can use[br]this and a little bit of

0:07:04.420,0:07:08.620
trigonometry to find the[br]relationship between a and r.

0:07:08.620,0:07:12.050
Because if we're able to solve[br]for a using r, then we can then

0:07:12.050,0:07:14.640
put that value of a in here and[br]we'll get the area

0:07:14.640,0:07:15.690
of our triangle.

0:07:15.690,0:07:17.600
And then we could subtract[br]that from the area of the

0:07:17.600,0:07:20.070
circle, and we're done.

0:07:20.070,0:07:22.050
We will have solved[br]the problem.

0:07:22.050,0:07:24.610
So let's see if we can do that.

0:07:24.610,0:07:29.340
So we have an angle[br]here of 60 degrees.

0:07:29.340,0:07:32.050
Half of this whole central[br]angle right there.

0:07:32.050,0:07:35.890
If this angle is 60 degrees,[br]we have a/2 that's

0:07:35.890,0:07:37.380
opposite to this angle.

0:07:37.380,0:07:43.480
So we have an opposite[br]is equal to a/2.

0:07:43.480,0:07:44.860
And we also have[br]the hypotenuse.

0:07:44.860,0:07:45.040
Right?

0:07:45.040,0:07:46.870
This is a right[br]triangle right here.

0:07:46.870,0:07:49.830
You're just going straight[br]down, and you're bisecting

0:07:49.830,0:07:50.840
that opposite side.

0:07:50.840,0:07:52.640
This is a right triangle.

0:07:52.640,0:07:54.000
So we can do a little[br]trigonometry.

0:07:54.000,0:08:02.550
Our opposite is a/2, the[br]hypotenuse is equal to r.

0:08:02.550,0:08:05.020
This is the hypotenuse, right[br]here, of our right triangle.

0:08:05.020,0:08:06.360
So that is equal to 2.

0:08:06.360,0:08:12.440
So what trig ratio is the[br]ratio of an angle's opposite

0:08:12.440,0:08:14.920
side to hypotenuse?

0:08:14.920,0:08:18.910
So some of you all might get[br]tired of me doing this all

0:08:18.910,0:08:22.030
the time, but SOH CAH TOA.

0:08:22.030,0:08:27.070
SOH-- sin of an angle is[br]equal to the opposite

0:08:27.070,0:08:28.620
over the hypotenuse.

0:08:28.620,0:08:29.580
So let me scroll[br]down a little bit.

0:08:29.580,0:08:31.270
I'm running out of space.

0:08:31.270,0:08:38.700
So the sin of this angle right[br]here, the sin of 60 degrees, is

0:08:38.700,0:08:42.070
going to be equal to the[br]opposite side, is going to be

0:08:42.070,0:08:45.800
equal to a/2, over the[br]hypotenuse, which is

0:08:45.800,0:08:48.140
our radius-- over 2.

0:08:48.140,0:08:54.510
Which is equal to a/2[br]divided by 2 is a/4.

0:08:54.510,0:08:56.880
And what is sin of 60 degrees?

0:08:56.880,0:08:59.720
And if the word "sin" looks[br]completely foreign to you,

0:08:59.720,0:09:04.150
watch the first several videos[br]on the trigonometry playlist.

0:09:04.150,0:09:06.240
It shouldn't be too daunting.

0:09:06.240,0:09:08.310
sin of 60 degrees you[br]might remember from your

0:09:08.310,0:09:10.680
30-60-90 triangles.

0:09:10.680,0:09:13.210
So let me draw one right there.

0:09:13.210,0:09:15.705
So that is a 30-60-90 triangle.

0:09:15.705,0:09:21.540
If this is 60 degrees, that[br]is 30 degrees, that is 90.

0:09:21.540,0:09:26.660
You might remember that this is[br]of length 1, this is going to

0:09:26.660,0:09:29.520
be of length 1/2, and this[br]is going to be of length

0:09:29.520,0:09:31.370
square root of 3 over 2.

0:09:31.370,0:09:35.300
So the sin of 60 degrees is[br]opposite over hypotenuse.

0:09:35.300,0:09:37.770
Square root of 3 over 2 over 1.

0:09:37.770,0:09:40.940
sin of 60 degrees.

0:09:40.940,0:09:42.840
If you don't have a calculator,[br]you could just use this--

0:09:42.840,0:09:44.930
is square root of 3 over 2.

0:09:44.930,0:09:48.890
So this right here is[br]square root of 3 over 2.

0:09:48.890,0:09:51.280
Now we can solve for a.

0:09:51.280,0:09:56.920
Square root of 3 over[br]2 is equal to a/4.

0:09:56.920,0:09:59.610
Let's multiply both sides by 4.

0:09:59.610,0:10:01.640
So you get this 4 cancels out.

0:10:01.640,0:10:03.445
You multiply 4 here.

0:10:03.445,0:10:04.480
This becomes a 2.

0:10:04.480,0:10:05.660
This becomes a 1.

0:10:05.660,0:10:09.250
You get a is equal to[br]2 square roots of 3.

0:10:09.250,0:10:11.000
We're in the home stretch.

0:10:11.000,0:10:15.420
We just figured out the length[br]of each of these sides.

0:10:15.420,0:10:17.460
We used Heron's formula to[br]figure out the area of the

0:10:17.460,0:10:19.080
triangle in terms[br]of those lengths.

0:10:19.080,0:10:22.110
So we just substitute this[br]value of a into there

0:10:22.110,0:10:24.670
to get our actual area.

0:10:24.670,0:10:30.380
So our triangle's area[br]is equal to a squared.

0:10:30.380,0:10:31.690
What's a squared?

0:10:31.690,0:10:37.780
That is 2 square roots[br]of 3 squared, times the

0:10:37.780,0:10:42.710
square root of 3 over 4.

0:10:42.710,0:10:45.470
We just did a squared times[br]the square root of 3 over 4.

0:10:45.470,0:10:51.950
This is going to be equal[br]to 4 times 3 times the

0:10:51.950,0:10:53.930
square of 3 over 4.

0:10:53.930,0:10:55.350
These 4's cancel.

0:10:55.350,0:10:58.360
So the area of our triangle[br]we got is 3 times the

0:10:58.360,0:11:00.770
square root of 3.

0:11:00.770,0:11:03.160
So the area here is 3[br]square roots of 3.

0:11:03.160,0:11:06.480
That's the area of[br]this entire triangle.

0:11:06.480,0:11:08.810
Now, to go back to what this[br]question was all about.

0:11:08.810,0:11:12.970
The area of this orange area[br]outside of the triangle

0:11:12.970,0:11:14.530
and inside of the circle.

0:11:14.530,0:11:18.460
Well, the area of[br]our circle is 4 pi.

0:11:18.460,0:11:23.340
And from that we subtract[br]the area of the triangle,

0:11:23.340,0:11:25.170
3 square roots of 3.

0:11:25.170,0:11:27.390
And we are done.

0:11:27.390,0:11:28.670
This is our answer.

0:11:28.670,0:11:35.270
This is the area of this[br]orange region right there.

0:11:35.270,0:11:37.800
Anyway, hopefully[br]you found that fun.

0:11:37.800,0:11:38.044