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So we've got a circle here--
doesn't look like a perfect

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00:00:05,610 --> 00:00:09,690
circle, but we can use our
imaginations --and let's say

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00:00:09,690 --> 00:00:21,780
it's got a radius of 3 meters.

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00:00:21,780 --> 00:00:24,580
My question, or the question
we're going to answer in this

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00:00:24,580 --> 00:00:29,020
video is what is the
area of this circle?

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00:00:29,020 --> 00:00:31,900
And remember, the area is just
how much space this circle

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takes up on a surface, or on
this computer screen that

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00:00:34,760 --> 00:00:36,285
you're watching, or on
this piece of paper.

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00:00:36,285 --> 00:00:39,330


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00:00:39,330 --> 00:00:42,100
If this was a room, it's how
much carpeting you would need

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00:00:42,100 --> 00:00:44,240
to fill out this circular room.

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That's what the area is.

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00:00:45,800 --> 00:00:48,860


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00:00:48,860 --> 00:00:51,370
Now, I'm not going to prove it
to you, and we'll do that

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00:00:51,370 --> 00:00:54,950
later, but the area for circle
just takes on a fairly

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00:00:54,950 --> 00:00:58,520
straightforward formula and I
want to just get you used to

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00:00:58,520 --> 00:00:59,610
applying that formula.

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So the area of a circle
is equal to pi.

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00:01:04,860 --> 00:01:08,710
Remember, pi was that number
that people figured out was the

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ratio between the circumference
and the diameter of the circle.

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It's 3.14159, keeps
going on and on and on.

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It's just the number, but
it's a very magical number.

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Pi times the radius squared.

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00:01:19,765 --> 00:01:22,600


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00:01:22,600 --> 00:01:26,810
In fact another way of defining
pi:-- you could even rewrite

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this right here --the area over
your radius squared- so

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this is your radius.

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If you multiply the radius
times itself you could imagine

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that would be the area of a
cube that's like that --that

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the ratio between the area of
this entire circle and the

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00:01:44,180 --> 00:01:49,235
ratio of this cube right
here-- or this square.

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I shouldn't say a cube.

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Cube would be if we went into
3D --but the ratio of the area

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00:01:54,640 --> 00:01:59,260
of the circle to this square
right here is also

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equal to pie.

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That could be actually
an alternate way of

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00:02:01,940 --> 00:02:03,730
defining what pi is.

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00:02:03,730 --> 00:02:06,790
And if you were to measure it
very carefully using-- there's

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00:02:06,790 --> 00:02:10,230
thousands of methods you could
do it --you would get 3.14159

40
00:02:10,230 --> 00:02:12,550
and keep going on
and on and on.

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00:02:12,550 --> 00:02:14,340
But we're not going to delve
too deeply into that.

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00:02:14,340 --> 00:02:17,100
Maybe one day I'll make a
whole play list on pi.

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But we just need to know that
the area is equal to pi times

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00:02:19,010 --> 00:02:21,560
r squared, so let's just
apply the numbers here.

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00:02:21,560 --> 00:02:27,595
So in our example, the area is
equal to pi times 3 meters

46
00:02:27,595 --> 00:02:34,360
squared, which is equal to pi
times 9 meters squared, or the

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00:02:34,360 --> 00:02:38,100
conventional way to write this
is, equal to 9pi

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00:02:38,100 --> 00:02:39,770
meters squared.

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00:02:39,770 --> 00:02:43,210
And remembered 9pi, the
convention is just to leave it

50
00:02:43,210 --> 00:02:48,640
that way, but this is the same
thing as 9 times 3.14159, which

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00:02:48,640 --> 00:02:52,140
is probably going to be, like,
28 point something

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00:02:52,140 --> 00:02:52,950
meters squared.

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00:02:52,950 --> 00:02:56,410
Just remember, this is just
some number and it's not 9.

54
00:02:56,410 --> 00:02:59,780
It's actually closer to 28
because it's going to be

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00:02:59,780 --> 00:03:04,050
9 times 3.14159, but we
just leave it like that.

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00:03:04,050 --> 00:03:06,310
And that normally will be
good enough for you to

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00:03:06,310 --> 00:03:08,660
say, hey that is my area.

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That's my area: 9pi.

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00:03:10,830 --> 00:03:15,510
Now let's go the other way:
let's say I have a circle and

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00:03:15,510 --> 00:03:20,560
let's say that someone would
say that the area

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is equal to 16pi.

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What is the diameter of
that circle going to be?

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00:03:32,770 --> 00:03:34,840
Well, we know that area
is equal to pi times

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00:03:34,840 --> 00:03:35,510
the radius squared.

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So at least let's
figure out the radius.

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00:03:37,980 --> 00:03:44,750
So the area, 16pi, is equal to
pi times our radius squared.

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I'm just applying this formula.

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00:03:47,070 --> 00:03:48,790
We're just going to keep
applying this formula

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00:03:48,790 --> 00:03:50,540
over and over again when
we're dealing with area.

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00:03:50,540 --> 00:03:54,760
So area, which we've been
told is 16pi, is equal to

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00:03:54,760 --> 00:03:57,400
pi times radius squared.

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00:03:57,400 --> 00:04:01,340
Now, if we divide both sides of
this equation by pi, we get

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00:04:01,340 --> 00:04:04,390
16 is equal to r squared.

74
00:04:04,390 --> 00:04:06,110
And then you take the square
root of both sides and

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00:04:06,110 --> 00:04:08,410
you get 4 is equal to r.

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00:04:08,410 --> 00:04:10,860
I guess r could also be equal
to negative 4, but we're

77
00:04:10,860 --> 00:04:12,780
dealing with distances here;
you can't have a

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00:04:12,780 --> 00:04:14,090
negative radius.

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00:04:14,090 --> 00:04:17,000
Or at least in the world
we're living in right now.

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00:04:17,000 --> 00:04:18,830
Just keep things simple;
we just want to keep our

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00:04:18,830 --> 00:04:19,900
distances positives.

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00:04:19,900 --> 00:04:23,820
So let's say that this
has a radius of 4.

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00:04:23,820 --> 00:04:26,550
Now if the radius is 4,
what is its diameter?

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00:04:26,550 --> 00:04:29,140
Well, the diameter is always
going to be 2 times the radius.

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00:04:29,140 --> 00:04:31,480
So this 4, we're going to
have another 4 there.

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00:04:31,480 --> 00:04:35,690
The diameter is equal to 8.

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00:04:35,690 --> 00:04:38,970
Now let's do a slightly harder
one that will kind of compound

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00:04:38,970 --> 00:04:42,060
some of the other things that
we've learned in the past.

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00:04:42,060 --> 00:04:45,400
So let's say that I
have a circle here.

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00:04:45,400 --> 00:04:59,650
Let's say that its
circumference is equal to 20pi,

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00:04:59,650 --> 00:05:01,430
and I want to know its area.

92
00:05:01,430 --> 00:05:04,590


93
00:05:04,590 --> 00:05:06,540
So way you do all these
problems is just figure out

94
00:05:06,540 --> 00:05:09,020
everything you can, given what
they've given you, and then

95
00:05:09,020 --> 00:05:12,010
maybe you can work out the
thing they're asking for.

96
00:05:12,010 --> 00:05:14,380
So if I know that the
circumference is 25, what do

97
00:05:14,380 --> 00:05:16,440
I know about its radius?

98
00:05:16,440 --> 00:05:19,280
Well, we saw in the last video
that the circumference is equal

99
00:05:19,280 --> 00:05:23,400
to 2pi times the radius.

100
00:05:23,400 --> 00:05:25,650
So if the circumference is
equal to 20pi, we could write

101
00:05:25,650 --> 00:05:30,530
that 20pi is the circumference
is equal to 2pi

102
00:05:30,530 --> 00:05:32,270
times the radius.

103
00:05:32,270 --> 00:05:36,600
Now if you divide both sides of
this by pi, those cancel out.

104
00:05:36,600 --> 00:05:39,830
Then if you divide both sides
by 2, this becomes a 1, this

105
00:05:39,830 --> 00:05:43,430
becomes a 10, or you get
the radius is equal to 10.

106
00:05:43,430 --> 00:05:44,600
Which makes sense, right?

107
00:05:44,600 --> 00:05:48,560
2pi times 10 is going
to be equal to 20pi.

108
00:05:48,560 --> 00:05:50,220
So we've figured
out our radius.

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00:05:50,220 --> 00:05:57,280
Now, we know that the area is
equal to pi times r squared.

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00:05:57,280 --> 00:05:59,860
And lucky for us, using the
circumference, we were able

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00:05:59,860 --> 00:06:01,420
to figure out the radius.

112
00:06:01,420 --> 00:06:04,690
Now using the radius, we
can figure out the area.

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00:06:04,690 --> 00:06:11,500
So the area is going to be
equal to pi times r squared,--

114
00:06:11,500 --> 00:06:17,040
r is 10 --times 10 squared,
which is equal to pi times 100.

115
00:06:17,040 --> 00:06:21,000
Or it's equal to 100pi.

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00:06:21,000 --> 00:06:21,780
Just like that.

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00:06:21,780 --> 00:06:25,350
so your circumference was 20pi,
when you went around the

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00:06:25,350 --> 00:06:29,910
circle, but your area of
your circle is 100pi.

119
00:06:29,910 --> 00:06:34,090
And if I gave you units it
would be 100pi units squared.

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00:06:34,090 --> 00:06:37,640
That is your area
right there: 100pi.

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00:06:37,640 --> 00:06:41,170
Anyway, I think that's pretty
good initial exposure to

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00:06:41,170 --> 00:06:42,360
the area of a circle.

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I'll see in the next video.