0:00:00.690,0:00:05.610
So we've got a circle here--[br]doesn't look like a perfect

0:00:05.610,0:00:09.690
circle, but we can use our[br]imaginations --and let's say

0:00:09.690,0:00:21.780
it's got a radius of 3 meters.

0:00:21.780,0:00:24.580
My question, or the question[br]we're going to answer in this

0:00:24.580,0:00:29.020
video is what is the[br]area of this circle?

0:00:29.020,0:00:31.900
And remember, the area is just[br]how much space this circle

0:00:31.900,0:00:34.760
takes up on a surface, or on[br]this computer screen that

0:00:34.760,0:00:36.285
you're watching, or on[br]this piece of paper.

0:00:36.285,0:00:39.330


0:00:39.330,0:00:42.100
If this was a room, it's how[br]much carpeting you would need

0:00:42.100,0:00:44.240
to fill out this circular room.

0:00:44.240,0:00:45.800
That's what the area is.

0:00:45.800,0:00:48.860


0:00:48.860,0:00:51.370
Now, I'm not going to prove it[br]to you, and we'll do that

0:00:51.370,0:00:54.950
later, but the area for circle[br]just takes on a fairly

0:00:54.950,0:00:58.520
straightforward formula and I[br]want to just get you used to

0:00:58.520,0:00:59.610
applying that formula.

0:00:59.610,0:01:04.860
So the area of a circle[br]is equal to pi.

0:01:04.860,0:01:08.710
Remember, pi was that number[br]that people figured out was the

0:01:08.710,0:01:11.750
ratio between the circumference[br]and the diameter of the circle.

0:01:11.750,0:01:15.420
It's 3.14159, keeps[br]going on and on and on.

0:01:15.420,0:01:17.880
It's just the number, but[br]it's a very magical number.

0:01:17.880,0:01:19.765
Pi times the radius squared.

0:01:19.765,0:01:22.600


0:01:22.600,0:01:26.810
In fact another way of defining[br]pi:-- you could even rewrite

0:01:26.810,0:01:32.690
this right here --the area over[br]your radius squared- so

0:01:32.690,0:01:33.650
this is your radius.

0:01:33.650,0:01:37.110
If you multiply the radius[br]times itself you could imagine

0:01:37.110,0:01:41.100
that would be the area of a[br]cube that's like that --that

0:01:41.100,0:01:44.180
the ratio between the area of[br]this entire circle and the

0:01:44.180,0:01:49.235
ratio of this cube right[br]here-- or this square.

0:01:49.235,0:01:50.330
I shouldn't say a cube.

0:01:50.330,0:01:54.640
Cube would be if we went into[br]3D --but the ratio of the area

0:01:54.640,0:01:59.260
of the circle to this square[br]right here is also

0:01:59.260,0:02:00.230
equal to pie.

0:02:00.230,0:02:01.940
That could be actually[br]an alternate way of

0:02:01.940,0:02:03.730
defining what pi is.

0:02:03.730,0:02:06.790
And if you were to measure it[br]very carefully using-- there's

0:02:06.790,0:02:10.230
thousands of methods you could[br]do it --you would get 3.14159

0:02:10.230,0:02:12.550
and keep going on[br]and on and on.

0:02:12.550,0:02:14.340
But we're not going to delve[br]too deeply into that.

0:02:14.340,0:02:17.100
Maybe one day I'll make a[br]whole play list on pi.

0:02:17.100,0:02:19.010
But we just need to know that[br]the area is equal to pi times

0:02:19.010,0:02:21.560
r squared, so let's just[br]apply the numbers here.

0:02:21.560,0:02:27.595
So in our example, the area is[br]equal to pi times 3 meters

0:02:27.595,0:02:34.360
squared, which is equal to pi[br]times 9 meters squared, or the

0:02:34.360,0:02:38.100
conventional way to write this[br]is, equal to 9pi

0:02:38.100,0:02:39.770
meters squared.

0:02:39.770,0:02:43.210
And remembered 9pi, the[br]convention is just to leave it

0:02:43.210,0:02:48.640
that way, but this is the same[br]thing as 9 times 3.14159, which

0:02:48.640,0:02:52.140
is probably going to be, like,[br]28 point something

0:02:52.140,0:02:52.950
meters squared.

0:02:52.950,0:02:56.410
Just remember, this is just[br]some number and it's not 9.

0:02:56.410,0:02:59.780
It's actually closer to 28[br]because it's going to be

0:02:59.780,0:03:04.050
9 times 3.14159, but we[br]just leave it like that.

0:03:04.050,0:03:06.310
And that normally will be[br]good enough for you to

0:03:06.310,0:03:08.660
say, hey that is my area.

0:03:08.660,0:03:10.830
That's my area: 9pi.

0:03:10.830,0:03:15.510
Now let's go the other way:[br]let's say I have a circle and

0:03:15.510,0:03:20.560
let's say that someone would[br]say that the area

0:03:20.560,0:03:24.080
is equal to 16pi.

0:03:24.080,0:03:32.770
What is the diameter of[br]that circle going to be?

0:03:32.770,0:03:34.840
Well, we know that area[br]is equal to pi times

0:03:34.840,0:03:35.510
the radius squared.

0:03:35.510,0:03:37.980
So at least let's[br]figure out the radius.

0:03:37.980,0:03:44.750
So the area, 16pi, is equal to[br]pi times our radius squared.

0:03:44.750,0:03:47.070
I'm just applying this formula.

0:03:47.070,0:03:48.790
We're just going to keep[br]applying this formula

0:03:48.790,0:03:50.540
over and over again when[br]we're dealing with area.

0:03:50.540,0:03:54.760
So area, which we've been[br]told is 16pi, is equal to

0:03:54.760,0:03:57.400
pi times radius squared.

0:03:57.400,0:04:01.340
Now, if we divide both sides of[br]this equation by pi, we get

0:04:01.340,0:04:04.390
16 is equal to r squared.

0:04:04.390,0:04:06.110
And then you take the square[br]root of both sides and

0:04:06.110,0:04:08.410
you get 4 is equal to r.

0:04:08.410,0:04:10.860
I guess r could also be equal[br]to negative 4, but we're

0:04:10.860,0:04:12.780
dealing with distances here;[br]you can't have a

0:04:12.780,0:04:14.090
negative radius.

0:04:14.090,0:04:17.000
Or at least in the world[br]we're living in right now.

0:04:17.000,0:04:18.830
Just keep things simple;[br]we just want to keep our

0:04:18.830,0:04:19.900
distances positives.

0:04:19.900,0:04:23.820
So let's say that this[br]has a radius of 4.

0:04:23.820,0:04:26.550
Now if the radius is 4,[br]what is its diameter?

0:04:26.550,0:04:29.140
Well, the diameter is always[br]going to be 2 times the radius.

0:04:29.140,0:04:31.480
So this 4, we're going to[br]have another 4 there.

0:04:31.480,0:04:35.690
The diameter is equal to 8.

0:04:35.690,0:04:38.970
Now let's do a slightly harder[br]one that will kind of compound

0:04:38.970,0:04:42.060
some of the other things that[br]we've learned in the past.

0:04:42.060,0:04:45.400
So let's say that I[br]have a circle here.

0:04:45.400,0:04:59.650
Let's say that its[br]circumference is equal to 20pi,

0:04:59.650,0:05:01.430
and I want to know its area.

0:05:01.430,0:05:04.590


0:05:04.590,0:05:06.540
So way you do all these[br]problems is just figure out

0:05:06.540,0:05:09.020
everything you can, given what[br]they've given you, and then

0:05:09.020,0:05:12.010
maybe you can work out the[br]thing they're asking for.

0:05:12.010,0:05:14.380
So if I know that the[br]circumference is 25, what do

0:05:14.380,0:05:16.440
I know about its radius?

0:05:16.440,0:05:19.280
Well, we saw in the last video[br]that the circumference is equal

0:05:19.280,0:05:23.400
to 2pi times the radius.

0:05:23.400,0:05:25.650
So if the circumference is[br]equal to 20pi, we could write

0:05:25.650,0:05:30.530
that 20pi is the circumference[br]is equal to 2pi

0:05:30.530,0:05:32.270
times the radius.

0:05:32.270,0:05:36.600
Now if you divide both sides of[br]this by pi, those cancel out.

0:05:36.600,0:05:39.830
Then if you divide both sides[br]by 2, this becomes a 1, this

0:05:39.830,0:05:43.430
becomes a 10, or you get[br]the radius is equal to 10.

0:05:43.430,0:05:44.600
Which makes sense, right?

0:05:44.600,0:05:48.560
2pi times 10 is going[br]to be equal to 20pi.

0:05:48.560,0:05:50.220
So we've figured[br]out our radius.

0:05:50.220,0:05:57.280
Now, we know that the area is[br]equal to pi times r squared.

0:05:57.280,0:05:59.860
And lucky for us, using the[br]circumference, we were able

0:05:59.860,0:06:01.420
to figure out the radius.

0:06:01.420,0:06:04.690
Now using the radius, we[br]can figure out the area.

0:06:04.690,0:06:11.500
So the area is going to be[br]equal to pi times r squared,--

0:06:11.500,0:06:17.040
r is 10 --times 10 squared,[br]which is equal to pi times 100.

0:06:17.040,0:06:21.000
Or it's equal to 100pi.

0:06:21.000,0:06:21.780
Just like that.

0:06:21.780,0:06:25.350
so your circumference was 20pi,[br]when you went around the

0:06:25.350,0:06:29.910
circle, but your area of[br]your circle is 100pi.

0:06:29.910,0:06:34.090
And if I gave you units it[br]would be 100pi units squared.

0:06:34.090,0:06:37.640
That is your area[br]right there: 100pi.

0:06:37.640,0:06:41.170
Anyway, I think that's pretty[br]good initial exposure to

0:06:41.170,0:06:42.360
the area of a circle.

0:06:42.360,0:06:44.450
I'll see in the next video.