WEBVTT

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The circle is arguably the most
fundamental shape in our

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universe, whether you look at
the shapes of orbits of

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planets, whether you look at
wheels, whether you look at

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things on kind of a
molecular level.

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The circle just keeps
showing up over and

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over and over again.

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So it's probably worthwhile for
us to understand some of the

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properties of the circle.

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So the first thing when people
kind of discovered the circle,

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and you just have a look at the
moon to see a circle, but the

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first time they said well, what
are the properties

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of any circle?

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So the first one they might
want to say is well, a circle

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is all of the points that are
equal distant from the

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center of the circle.

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All of these points along the
edge are equal distant from

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that center right there.

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So one of the first things
someone might want to ask is

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what is that distance, that
equal distance that everything

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is from the center?

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Right there.

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We call that the
radius of the circle.

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It's just the distance from
the center out to the edge.

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If that radius is 3
centimeters, then this radius

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is going to be 3 centimeters.

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And this radius is going
to be 3 centimeters.

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It's never going to change.

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By definition, a circle is all
of the points that are equal

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distant from the center point.

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And that distance
is the radius.

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Now the next most interesting
thing about that, people might

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say well, how fat
is the circle?

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How wide is it along
its widest point?

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Or if you just want to cut it
along its widest point, what

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is that distance right there?

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And it doesn't have to be just
right there, I could have just

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as easily cut it along its
widest point right there.

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I just wouldn't be cutting it
like some place like that

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because that wouldn't be
along its widest point.

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There's multiple places
where I could cut it

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along its widest point.

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Well, we just saw the radius
and we see that widest point

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goes through the center
and just keeps going.

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So it's essentially two radii.

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You got one radius there
and then you have another

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radius over there.

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We call this distance along
the widest point of the

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circle, the diameter.

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So that is the diameter
of the circle.

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It has a very easy
relationship with the radius.

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The diameter is equal to
two times the radius.

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Now, the next most interesting
thing that you might be

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wondering about a circle is how
far is it around the circle?

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So if you were to get your tape
measure out and you were to

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measure around the circle like
that, what's that distance?

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We call that word the
circumference of the circle.

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Now, we know how the diameter
and the radius relates, but how

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does the circumference relate
to, say, the diameter.

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And if you're not really used
to the diameter, it's very

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easy to figure out how it
relates to the radius.

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Well, many thousands of years
ago, people took their tape

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measures out and they keep
measuring circumferences

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and radiuses.

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And let's say when their tape
measures weren't so good,

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let's say they measured the
circumference of the circle

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and they would get well, it
looks like it's about 3.

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And then they measure the
radius of the circle right here

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or the diameter of that circle,
and they'd say oh, the diameter

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looks like it's about 1.

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So they would say -- let
me write this down.

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So we're worried about
the ratio -- let me

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write it like this.

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The ratio of the circumference
to the diameter.

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So let's say that somebody had
some circle over here -- let's

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say they had this circle, and
the first time with not that

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good of a tape measure, they
measured around the circle

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and they said hey, it's
roughly equal to 3 meters

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when I go around it.

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And when I measure the
diameter of the circle,

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it's roughly equal to 1.

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OK, that's interesting.

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Maybe the ratio of
the circumference of

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the diameter's 3.

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So maybe the circumference
is always three

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times the diameter.

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Well that was just for this
circle, but let's say they

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measured some other
circle here.

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It's like this -- I
drew it smaller.

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Let's say that on this circle
they measured around it and

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they found out that the
circumference is 6 centimeters,

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roughly -- we have a bad
tape measure right then.

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Then they find out
that the diameter is

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roughly 2 centimeters.

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And once again, the ratio of
the circumference of the

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diameter was roughly 3.

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OK, this is a neat
property of circles.

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Maybe the ratio of the
circumference to the diameters

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always fixed for any circle.

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So they said let me
study this further.

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So they got better
tape measures.

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When they got better tape
measures, they measured hey,

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my diameter's definitely 1.

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They say my diameter's
definitely 1, but when I

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measure my circumference
a little bit, I realize

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it's closer to 3.1.

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And the same thing
with this over here.

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They notice that this
ratio is closer to 3.1.

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Then they kept measuring it
better and better and better,

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and then they realized that
they were getting this number,

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they just kept measuring it
better and better and they were

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getting this number 3.14159.

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And they just kept adding
digits and it would

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never repeat.

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It was a strange fascinating
metaphysical number

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that kept showing up.

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So since this number was so
fundamental to our universe,

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because the circle is so
fundamental to our universe,

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and it just showed up
for every circle.

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The ratio of the circumference
of the diameter was this

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kind of magical number,
they gave it a name.

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They called it pi, or you could
just give it the Latin or the

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Greek letter pi --
just like that.

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That represents this number
which is arguably the most

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fascinating number
in our universe.

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It first shows up as the ratio
of the circumference to the

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diameter, but you're going to
learn as you go through your

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mathematical journey, that
it shows up everywhere.

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It's one of these fundamental
things about the universe that

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just makes you think that
there's some order to it.

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But anyway, how can we
use this in I guess

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our basic mathematics?

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So we know, or I'm telling you,
that the ratio of the

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circumference to the diameter
-- when I say the ratio,

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literally I'm just saying if
you divide the circumference by

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the diameter, you're
going to get pi.

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Pi is just this number.

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I could write 3.14159 and just
keep going on and on and on,

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but that would be a waste of
space and it would just be hard

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to deal with, so people just
write this Greek

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letter pi there.

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So, how can we relate this?

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We can multiply both sides of
this by the diameter and we

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could say that the
circumference is equal to pi

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times the diameter.

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Or since the diameter is equal
to 2 times the radius, we could

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say that the circumference is
equal to pi times 2

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times the radius.

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Or the form that you're
most likely to see it,

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it's equal to 2 pi r.

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So let's see if we can apply
that to some problems.

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So let's say I have a circle
just like that, and I were to

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tell you it has a radius --
it's radius right there is 3.

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So, 3 -- let me write this down
-- so the radius is equal to 3.

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Maybe it's 3 meters --
put some units in there.

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What is the circumference
of the circle?

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The circumference is equal to
2 times pi times the radius.

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So it's going to be equal to 2
times pi times the radius,

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times 3 meters, which is
equal to 6 meters times

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pi or 6 pi meters.

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6 pi meters.

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Now I could multiply this out.

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Remember pi is just a number.

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Pi is 3.14159 going
on and on and on.

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So if I multiply 6 times that,
maybe I'll get 18 point

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something something something.

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If you have your calculator you
might want to do it, but for

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simplicity people just tend to
leave our numbers

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in terms of pi.

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Now I don't know what this is
if you multiply 6 times

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3.14159, I don't know if you
get something close to 19 or

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18, maybe it's approximately
18 point something

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something something.

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I don't have my calculator
in front of me.

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But instead of writing
that number, you just

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write 6 pi there.

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Actually, I think it
wouldn't quite cross the

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threshold to 19 yet.

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Now, let's ask
another question.

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What is the diameter
of the circle?

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Well if this radius is 3, the
diameter is just twice that.

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So it's just going to be 3
times 2 or 3 plus 3, which

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is equal to 6 meters.

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So the circumference is 6 pi
meters, the diameter is 6

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meters, the radius is 3 meters.

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Now let's go the other way.

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Let's say I have
another circle.

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Let's say I have
another circle here.

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And I were to tell you that
its circumference is equal

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to 10 meters -- that's the
circumference of the circle.

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If you were to put a tape
measure to go around it and

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someone were to ask you what is
the diameter of the circle?

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Well, we know that the diameter
times pi, we know that pi times

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the diameter is equal to
the circumference; is

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equal to 10 meters.

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So to solve for this we would
just divide both sides

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of this equation by pi.

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The diameter would equal
10 meters over pi or

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10 over pi meters.

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And that is just a number.

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If you have your calculator,
you could actually divide 10

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divided by 3.14159, you're
going to get 3 point something

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something something meters.

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I can't do it in my head.

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But this is just a number.

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But for simplicity we often
just leave it that way.

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Now what is the radius?

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Well, the radius is equal
to 1/2 the diameter.

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So this whole distance right
here is 10 over pi meters.

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If we just 1/2 of that, if
we just want the radius, we

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just multiply it times 1/2.

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So you have 1/2 times 10 over
pi, which is equal to 1/2 times

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10, or you just divide the
numerator and the

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denominator by 2.

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You get 5 there, so
you get 5 over pi.

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So the radius over
here is 5 over pi.

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Nothing super fancy about this.

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I think the thing that confuses
people the most is to just

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realize that pi is a number.

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Pi is just 3.14159 and it just
keeps going on and on and on.

00:10:38.640 --> 00:10:41.950
There's actually thousands of
books written about pi, so

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it's not like -- I don't know
if there's thousands, I'm

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exaggerating, but you could
write books about this number.

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But it's just a number.

00:10:49.340 --> 00:10:52.480
It's a very special number, and
if you wanted to write it in a

00:10:52.480 --> 00:10:54.390
way that you're used to writing
numbers, you could literally

00:10:54.390 --> 00:10:55.680
just multiply this out.

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But most the time people just
realize they like leaving

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things in terms of pi.

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Anyway, I'll leave you there.

00:11:01.680 --> 00:11:05.090
In the next video we'll figure
out the area of a circle.