WEBVTT

00:00:00.690 --> 00:00:05.610
So we've got a circle here--
doesn't look like a perfect

00:00:05.610 --> 00:00:09.690
circle, but we can use our
imaginations --and let's say

00:00:09.690 --> 00:00:21.780
it's got a radius of 3 meters.

00:00:21.780 --> 00:00:24.580
My question, or the question
we're going to answer in this

00:00:24.580 --> 00:00:29.020
video is what is the
area of this circle?

00:00:29.020 --> 00:00:31.900
And remember, the area is just
how much space this circle

00:00:31.900 --> 00:00:34.760
takes up on a surface, or on
this computer screen that

00:00:34.760 --> 00:00:36.285
you're watching, or on
this piece of paper.

00:00:36.285 --> 00:00:39.330


00:00:39.330 --> 00:00:42.100
If this was a room, it's how
much carpeting you would need

00:00:42.100 --> 00:00:44.240
to fill out this circular room.

00:00:44.240 --> 00:00:45.800
That's what the area is.

00:00:45.800 --> 00:00:48.860


00:00:48.860 --> 00:00:51.370
Now, I'm not going to prove it
to you, and we'll do that

00:00:51.370 --> 00:00:54.950
later, but the area for circle
just takes on a fairly

00:00:54.950 --> 00:00:58.520
straightforward formula and I
want to just get you used to

00:00:58.520 --> 00:00:59.610
applying that formula.

00:00:59.610 --> 00:01:04.860
So the area of a circle
is equal to pi.

00:01:04.860 --> 00:01:08.710
Remember, pi was that number
that people figured out was the

00:01:08.710 --> 00:01:11.750
ratio between the circumference
and the diameter of the circle.

00:01:11.750 --> 00:01:15.420
It's 3.14159, keeps
going on and on and on.

00:01:15.420 --> 00:01:17.880
It's just the number, but
it's a very magical number.

00:01:17.880 --> 00:01:19.765
Pi times the radius squared.

00:01:19.765 --> 00:01:22.600


00:01:22.600 --> 00:01:26.810
In fact another way of defining
pi:-- you could even rewrite

00:01:26.810 --> 00:01:32.690
this right here --the area over
your radius squared- so

00:01:32.690 --> 00:01:33.650
this is your radius.

00:01:33.650 --> 00:01:37.110
If you multiply the radius
times itself you could imagine

00:01:37.110 --> 00:01:41.100
that would be the area of a
cube that's like that --that

00:01:41.100 --> 00:01:44.180
the ratio between the area of
this entire circle and the

00:01:44.180 --> 00:01:49.235
ratio of this cube right
here-- or this square.

00:01:49.235 --> 00:01:50.330
I shouldn't say a cube.

00:01:50.330 --> 00:01:54.640
Cube would be if we went into
3D --but the ratio of the area

00:01:54.640 --> 00:01:59.260
of the circle to this square
right here is also

00:01:59.260 --> 00:02:00.230
equal to pie.

00:02:00.230 --> 00:02:01.940
That could be actually
an alternate way of

00:02:01.940 --> 00:02:03.730
defining what pi is.

00:02:03.730 --> 00:02:06.790
And if you were to measure it
very carefully using-- there's

00:02:06.790 --> 00:02:10.230
thousands of methods you could
do it --you would get 3.14159

00:02:10.230 --> 00:02:12.550
and keep going on
and on and on.

00:02:12.550 --> 00:02:14.340
But we're not going to delve
too deeply into that.

00:02:14.340 --> 00:02:17.100
Maybe one day I'll make a
whole play list on pi.

00:02:17.100 --> 00:02:19.010
But we just need to know that
the area is equal to pi times

00:02:19.010 --> 00:02:21.560
r squared, so let's just
apply the numbers here.

00:02:21.560 --> 00:02:27.595
So in our example, the area is
equal to pi times 3 meters

00:02:27.595 --> 00:02:34.360
squared, which is equal to pi
times 9 meters squared, or the

00:02:34.360 --> 00:02:38.100
conventional way to write this
is, equal to 9pi

00:02:38.100 --> 00:02:39.770
meters squared.

00:02:39.770 --> 00:02:43.210
And remembered 9pi, the
convention is just to leave it

00:02:43.210 --> 00:02:48.640
that way, but this is the same
thing as 9 times 3.14159, which

00:02:48.640 --> 00:02:52.140
is probably going to be, like,
28 point something

00:02:52.140 --> 00:02:52.950
meters squared.

00:02:52.950 --> 00:02:56.410
Just remember, this is just
some number and it's not 9.

00:02:56.410 --> 00:02:59.780
It's actually closer to 28
because it's going to be

00:02:59.780 --> 00:03:04.050
9 times 3.14159, but we
just leave it like that.

00:03:04.050 --> 00:03:06.310
And that normally will be
good enough for you to

00:03:06.310 --> 00:03:08.660
say, hey that is my area.

00:03:08.660 --> 00:03:10.830
That's my area: 9pi.

00:03:10.830 --> 00:03:15.510
Now let's go the other way:
let's say I have a circle and

00:03:15.510 --> 00:03:20.560
let's say that someone would
say that the area

00:03:20.560 --> 00:03:24.080
is equal to 16pi.

00:03:24.080 --> 00:03:32.770
What is the diameter of
that circle going to be?

00:03:32.770 --> 00:03:34.840
Well, we know that area
is equal to pi times

00:03:34.840 --> 00:03:35.510
the radius squared.

00:03:35.510 --> 00:03:37.980
So at least let's
figure out the radius.

00:03:37.980 --> 00:03:44.750
So the area, 16pi, is equal to
pi times our radius squared.

00:03:44.750 --> 00:03:47.070
I'm just applying this formula.

00:03:47.070 --> 00:03:48.790
We're just going to keep
applying this formula

00:03:48.790 --> 00:03:50.540
over and over again when
we're dealing with area.

00:03:50.540 --> 00:03:54.760
So area, which we've been
told is 16pi, is equal to

00:03:54.760 --> 00:03:57.400
pi times radius squared.

00:03:57.400 --> 00:04:01.340
Now, if we divide both sides of
this equation by pi, we get

00:04:01.340 --> 00:04:04.390
16 is equal to r squared.

00:04:04.390 --> 00:04:06.110
And then you take the square
root of both sides and

00:04:06.110 --> 00:04:08.410
you get 4 is equal to r.

00:04:08.410 --> 00:04:10.860
I guess r could also be equal
to negative 4, but we're

00:04:10.860 --> 00:04:12.780
dealing with distances here;
you can't have a

00:04:12.780 --> 00:04:14.090
negative radius.

00:04:14.090 --> 00:04:17.000
Or at least in the world
we're living in right now.

00:04:17.000 --> 00:04:18.830
Just keep things simple;
we just want to keep our

00:04:18.830 --> 00:04:19.900
distances positives.

00:04:19.900 --> 00:04:23.820
So let's say that this
has a radius of 4.

00:04:23.820 --> 00:04:26.550
Now if the radius is 4,
what is its diameter?

00:04:26.550 --> 00:04:29.140
Well, the diameter is always
going to be 2 times the radius.

00:04:29.140 --> 00:04:31.480
So this 4, we're going to
have another 4 there.

00:04:31.480 --> 00:04:35.690
The diameter is equal to 8.

00:04:35.690 --> 00:04:38.970
Now let's do a slightly harder
one that will kind of compound

00:04:38.970 --> 00:04:42.060
some of the other things that
we've learned in the past.

00:04:42.060 --> 00:04:45.400
So let's say that I
have a circle here.

00:04:45.400 --> 00:04:59.650
Let's say that its
circumference is equal to 20pi,

00:04:59.650 --> 00:05:01.430
and I want to know its area.

00:05:01.430 --> 00:05:04.590


00:05:04.590 --> 00:05:06.540
So way you do all these
problems is just figure out

00:05:06.540 --> 00:05:09.020
everything you can, given what
they've given you, and then

00:05:09.020 --> 00:05:12.010
maybe you can work out the
thing they're asking for.

00:05:12.010 --> 00:05:14.380
So if I know that the
circumference is 25, what do

00:05:14.380 --> 00:05:16.440
I know about its radius?

00:05:16.440 --> 00:05:19.280
Well, we saw in the last video
that the circumference is equal

00:05:19.280 --> 00:05:23.400
to 2pi times the radius.

00:05:23.400 --> 00:05:25.650
So if the circumference is
equal to 20pi, we could write

00:05:25.650 --> 00:05:30.530
that 20pi is the circumference
is equal to 2pi

00:05:30.530 --> 00:05:32.270
times the radius.

00:05:32.270 --> 00:05:36.600
Now if you divide both sides of
this by pi, those cancel out.

00:05:36.600 --> 00:05:39.830
Then if you divide both sides
by 2, this becomes a 1, this

00:05:39.830 --> 00:05:43.430
becomes a 10, or you get
the radius is equal to 10.

00:05:43.430 --> 00:05:44.600
Which makes sense, right?

00:05:44.600 --> 00:05:48.560
2pi times 10 is going
to be equal to 20pi.

00:05:48.560 --> 00:05:50.220
So we've figured
out our radius.

00:05:50.220 --> 00:05:57.280
Now, we know that the area is
equal to pi times r squared.

00:05:57.280 --> 00:05:59.860
And lucky for us, using the
circumference, we were able

00:05:59.860 --> 00:06:01.420
to figure out the radius.

00:06:01.420 --> 00:06:04.690
Now using the radius, we
can figure out the area.

00:06:04.690 --> 00:06:11.500
So the area is going to be
equal to pi times r squared,--

00:06:11.500 --> 00:06:17.040
r is 10 --times 10 squared,
which is equal to pi times 100.

00:06:17.040 --> 00:06:21.000
Or it's equal to 100pi.

00:06:21.000 --> 00:06:21.780
Just like that.

00:06:21.780 --> 00:06:25.350
so your circumference was 20pi,
when you went around the

00:06:25.350 --> 00:06:29.910
circle, but your area of
your circle is 100pi.

00:06:29.910 --> 00:06:34.090
And if I gave you units it
would be 100pi units squared.

00:06:34.090 --> 00:06:37.640
That is your area
right there: 100pi.

00:06:37.640 --> 00:06:41.170
Anyway, I think that's pretty
good initial exposure to

00:06:41.170 --> 00:06:42.360
the area of a circle.

00:06:42.360 --> 00:06:44.450
I'll see in the next video.