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## Volume and surface area

# Cylinder volume & surface area

CCSS.Math: ,

## Video transcript

Let's find the volume of
a few more solid figures and then if we
have time, we might be able to do some
surface area problems. So let me draw a
cylinder over here. So that is the top
of my cylinder. And then this is the
height of my cylinder. This is the bottom
right over here. If this was
transparent, maybe you could see the back
side of the cylinder. So you could imagine this
kind of looks like a soda can. And let's say that the height of
my cylinder, h, is equal to 8. I'll give it some units. 8 centimeters,
that is my height. And then let's say
that the radius of one of the top of my
cylinder of my soda can, let's say that this radius over
here is equal to 4 centimeters. So what is the volume here? What is the volume going to be? And the idea here is
really the exact same thing that we saw in some of
the previous problems. If you can find the
surface area of one side and then figure out
how deep it goes, you'll be able to
figure out the volume. So what we're going to
do here is figure out the surface area of the
top of this cylinder, or the top of the soda can. And then we're going to
multiply it by its height. And that'll give us a volume. This will tell us
essentially, how many square centimeters
fit in this top. And then if we multiply
that by how many centimeters we go down, then that'll
give us the number of cubic centimeters in
this cylinder or soda can. So how do we figure
out this area up here? Well the area at the
top, this is just finding the area of a circle. You could imagine
drawing it like this. If we were to just
look at it straight on, that's a circle with a
radius of 4 centimeters. The area of a circle with a
radius 4 centimeters, area is equal to pi r squared. So it's going to be pi
times the radius squared, times 4 centimeters squared,
which is equal to 4 squared, is 16, times pi. And our units now are going
to be centimeters squared. Or another way to think of
these are square centimeters. So that's the area. The volume is going to be
this area times the height. So the volume is going
to be equal to 16 pi centimeters squared, times the
height, times 8 centimeters. And so, when you
do multiplication, the associative
property, you can kind of rearrange these things, and
the commutative property. It doesn't matter
what order you do it if it's all multiplication. So this is the same
thing as 16 times 8. Let's see. 8 times 8 is 64. 16 times 8 is twice that. So it's going to be 128 pi. Now you have centimeter
squared times centimeters. So that gives us
centimeters cubed. Or 128 pi cubic centimeters. Remember, pi is just a number. We write it as pi,
because it's kind of a crazy, irrational number,
that if you were to write it, you could never
completely write pi. 3.14159 keeps going
on, never a repeat. So we just leave it as pi. But if you wanted to figure it
out, you can get a calculator. And this would be 3.14
roughly, times 128. So it would be close to
400 cubic centimeters. Now, how would we
find the surface area? How would we find the surface
area of this figure over here? Well, part of the surface area
of the two surfaces, the top and the bottom. So that would be part
of the surface area. And then the bottom
over here would also be part of the surface area. So if we're trying to
find the surface area, it's definitely going to have
both of these areas here. So it's going to have the 16
pi centimeters squared twice. This is 16 pi. This is 16 pi
square centimeters. So it's going to have 2 times
16 pi centimeters squared. I'll keep the units still. So that covers the top and
the bottom of our soda can. And now we have to
figure out the surface area of this thing
that goes around. And the way I imagine
it is, imagine if you're trying to wrap this
thing with wrapping paper. So let me just draw a
little dotted line here. So imagine if you were
to cut it just like that. Cut the side of the soda can. And if you were to
unwind this thing that goes around it,
what would you have. Well, you would have something. You would end up
with a sheet of paper where this length
right over here is the same thing as
this length over here. And then it would be
completely unwound. And then these two
ends-- let me do it in magenta-- these two ends
used to touch each other. And-- I'm going to do it in a
color that I haven't used yet, I'll do it in pink--
these two ends used to touch each other when
it was all rolled together. And they used to touch each
other right over there. So the length of this
side and that side is going to be the same thing
as the height of my cylinder. So this is going to
be 8 centimeters. And then this over here is
also going to be 8 centimeters. And so the question
we need to ask ourselves is, what is
going to be this dimension right over here. And remember, that
dimension is essentially, how far did we go
around the cylinder. Well, if you think
about it, that's going to be the exact same
thing as the circumference of either the top or the
bottom of the cylinder. So what is the circumference? The circumference
of this circle right over here, which is the same
thing as the circumference of that circle over there, it
is 2 times the radius times pi. Or 2 pi times the radius. 2 pi times 4 centimeters, which
is equal to 8 pi centimeters. So this distance right over
here is the circumference of either the top or the
bottom of the cylinder. It's going to be
8 pi centimeters. So if you want to
find the surface area of just the wrapping,
just the part that goes around the cylinder,
not the top or the bottom, when you unwind it, it's going
to look like this rectangle. And so its area, the
area of just that part, is going to be equal to
8 centimeters times 8 pi centimeters. So let me do it this way. It's going to be 8 centimeters
times 8 pi centimeters. And that's equal to 64 pi. 8 times 8 is 64. You have your pi
centimeters squared. So when you want the surface
area of the whole thing, you have the top,
you have the bottom, we already threw those there. And then you want to find
the area of the thing around. We just figured that out. So it's going to be plus
64 pi centimeters squared. And now we just have
to calculate it. So this gives us 2 times 16
pi, is going to be equal to 32. That is 32 pi centimeters
squared, plus 64 pi. Let me scroll over to
the right a little bit. Plus 64 pi centimeters squared. And then 32 plus 64 is 96
pi centimeters squared. So it's equal to 96 pi
square centimeters, which is going to be a little bit
over 300 square centimeters. And notice, when we
did surface area, we got our answer in terms
of square centimeters. That makes sense,
because surface area, it's a two-dimensional
measurement. Think about how many
square centimeters can we fit on the surface
of the cylinder. When we did the volume,
we got centimeters cubed, or cubic centimeters. And that's because we're
trying to calculate how many one by one
by one centimeter cubes can we fit inside
of this structure. And so that's why it's
cubic centimeters. Anyway, hopefully that clarifies
things up a little bit.