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Current time:0:00Total duration:11:05

CCSS.Math:

The circle is arguably the most
fundamental shape in our universe, whether you look at
the shapes of orbits of planets, whether you look at
wheels, whether you look at things on kind of a
molecular level. The circle just keeps
showing up over and over and over again. So it's probably worthwhile for
us to understand some of the properties of the circle. So the first thing when people
kind of discovered the circle, and you just have a look at the
moon to see a circle, but the first time they said well, what
are the properties of any circle? So the first one they might
want to say is well, a circle is all of the points that are
equal distant from the center of the circle. All of these points along the
edge are equal distant from that center right there. So one of the first things
someone might want to ask is what is that distance, that
equal distance that everything is from the center? Right there. We call that the
radius of the circle. It's just the distance from
the center out to the edge. If that radius is 3
centimeters, then this radius is going to be 3 centimeters. And this radius is going
to be 3 centimeters. It's never going to change. By definition, a circle is all
of the points that are equal distant from the center point. And that distance
is the radius. Now the next most interesting
thing about that, people might say well, how fat
is the circle? How wide is it along
its widest point? Or if you just want to cut it
along its widest point, what is that distance right there? And it doesn't have to be just
right there, I could have just as easily cut it along its
widest point right there. I just wouldn't be cutting it
like some place like that because that wouldn't be
along its widest point. There's multiple places
where I could cut it along its widest point. Well, we just saw the radius
and we see that widest point goes through the center
and just keeps going. So it's essentially two radii. You got one radius there
and then you have another radius over there. We call this distance along
the widest point of the circle, the diameter. So that is the diameter
of the circle. It has a very easy
relationship with the radius. The diameter is equal to
two times the radius. Now, the next most interesting
thing that you might be wondering about a circle is how
far is it around the circle? So if you were to get your tape
measure out and you were to measure around the circle like
that, what's that distance? We call that word the
circumference of the circle. Now, we know how the diameter
and the radius relates, but how does the circumference relate
to, say, the diameter. And if you're not really used
to the diameter, it's very easy to figure out how it
relates to the radius. Well, many thousands of years
ago, people took their tape measures out and they keep
measuring circumferences and radiuses. And let's say when their tape
measures weren't so good, let's say they measured the
circumference of the circle and they would get well, it
looks like it's about 3. And then they measure the
radius of the circle right here or the diameter of that circle,
and they'd say oh, the diameter looks like it's about 1. So they would say -- let
me write this down. So we're worried about
the ratio -- let me write it like this. The ratio of the circumference
to the diameter. So let's say that somebody had
some circle over here -- let's say they had this circle, and
the first time with not that good of a tape measure, they
measured around the circle and they said hey, it's
roughly equal to 3 meters when I go around it. And when I measure the
diameter of the circle, it's roughly equal to 1. OK, that's interesting. Maybe the ratio of
the circumference of the diameter's 3. So maybe the circumference
is always three times the diameter. Well that was just for this
circle, but let's say they measured some other
circle here. It's like this -- I
drew it smaller. Let's say that on this circle
they measured around it and they found out that the
circumference is 6 centimeters, roughly -- we have a bad
tape measure right then. Then they find out
that the diameter is roughly 2 centimeters. And once again, the ratio of
the circumference of the diameter was roughly 3. OK, this is a neat
property of circles. Maybe the ratio of the
circumference to the diameters always fixed for any circle. So they said let me
study this further. So they got better
tape measures. When they got better tape
measures, they measured hey, my diameter's definitely 1. They say my diameter's
definitely 1, but when I measure my circumference
a little bit, I realize it's closer to 3.1. And the same thing
with this over here. They notice that this
ratio is closer to 3.1. Then they kept measuring it
better and better and better, and then they realized that
they were getting this number, they just kept measuring it
better and better and they were getting this number 3.14159. And they just kept adding
digits and it would never repeat. It was a strange fascinating
metaphysical number that kept showing up. So since this number was so
fundamental to our universe, because the circle is so
fundamental to our universe, and it just showed up
for every circle. The ratio of the circumference
of the diameter was this kind of magical number,
they gave it a name. They called it pi, or you could
just give it the Latin or the Greek letter pi --
just like that. That represents this number
which is arguably the most fascinating number
in our universe. It first shows up as the ratio
of the circumference to the diameter, but you're going to
learn as you go through your mathematical journey, that
it shows up everywhere. It's one of these fundamental
things about the universe that just makes you think that
there's some order to it. But anyway, how can we
use this in I guess our basic mathematics? So we know, or I'm telling you,
that the ratio of the circumference to the diameter
-- when I say the ratio, literally I'm just saying if
you divide the circumference by the diameter, you're
going to get pi. Pi is just this number. I could write 3.14159 and just
keep going on and on and on, but that would be a waste of
space and it would just be hard to deal with, so people just
write this Greek letter pi there. So, how can we relate this? We can multiply both sides of
this by the diameter and we could say that the
circumference is equal to pi times the diameter. Or since the diameter is equal
to 2 times the radius, we could say that the circumference is
equal to pi times 2 times the radius. Or the form that you're
most likely to see it, it's equal to 2 pi r. So let's see if we can apply
that to some problems. So let's say I have a circle
just like that, and I were to tell you it has a radius --
it's radius right there is 3. So, 3 -- let me write this down
-- so the radius is equal to 3. Maybe it's 3 meters --
put some units in there. What is the circumference
of the circle? The circumference is equal to
2 times 5 times the radius. So it's going to be equal to 2
times pi times the radius, times 3 meters, which is
equal to 6 meters times pi or 6 pi meters. 6 pi meters. Now I could multiply this out. Remember pi is just a number. Pi is 3.14159 going
on and on and on. So if I multiply 6 times that,
maybe I'll get 18 point something something something. If you have your calculator you
might want to do it, but for simplicity people just tend to
leave our numbers in terms of pi. Now I don't know what this is
if you multiply 6 times 3.14159, I don't know if you
get something close to 19 or 18, maybe it's approximately
18 point something something something. I don't have my calculator
in front of me. But instead of writing
that number, you just write 6 pi there. Actually, I think it
wouldn't quite cross the threshold to 19 yet. Now, let's ask
another question. What is the diameter
of the circle? Well if this radius is 3, the
diameter is just twice that. So it's just going to be 3
times 2 or 3 plus 3, which is equal to 6 meters. So the circumference is 6 pi
meters, the diameter is 6 meters, the radius is 3 meters. Now let's go the other way. Let's say I have
another circle. Let's say I have
another circle here. And I were to tell you that
its circumference is equal to 10 meters -- that's the
circumference of the circle. If you were to put a tape
measure to go around it and someone were to ask you what is
the diameter of the circle? Well, we know that the diameter
times pi, we know that pi times the diameter is equal to
the circumference; is equal to 10 meters. So to solve for this we would
just divide both sides of this equation by pi. The diameter would equal
10 meters over pi or 10 over pi meters. And that is just a number. If you have your calculator,
you could actually divide 10 divided by 3.14159, you're
going to get 3 point something something something meters. I can't do it in my head. But this is just a number. But for simplicity we often
just leave it that way. Now what is the radius? Well, the radius is equal
to 1/2 the diameter. So this whole distance right
here is 10 over pi meters. If we just 1/2 of that, if
we just want the radius, we just multiply it times 1/2. So you have 1/2 times 10 over
pi, which is equal to 1/2 times 10, or you just divide the
numerator and the denominator by 2. You get 5 there, so
you get 5 over pi. So the radius over
here is 5 over pi. Nothing super fancy about this. I think the thing that confuses
people the most is to just realize that pi is a number. Pi is just 3.14159 and it just
keeps going on and on and on. There's actually thousands of
books written about pi, so it's not like -- I don't know
if there's thousands, I'm exaggerating, but you could
write books about this number. But it's just a number. It's a very special number, and
if you wanted to write it in a way that you're used to writing
numbers, you could literally just multiply this out. But most the time people just
realize they like leaving things in terms of pi. Anyway, I'll leave you there. In the next video we'll figure
out the area of a circle.