Video Transcript
In this video, we’re going to be
talking about how to find the volume of a cylinder. First, we’re going to take a look
at prisms and how to work out the volume of a prism. And then, we’re going to explain
how a cylinder is a circular prism. Finally, we’ll look at a few
examples of cylinders and how to work out their volumes.
Before we talk about cylinders
then, let’s think about prisms. A prism is a 3D shape with a
constant cross section. For example, here’s a cuboid. I’ve marked in the cross section
with this orange stripey bit here. And if we were to cut this prism at
any point, for example, at this point, and then look at the slice that we get, then
we would still have exactly the same cross section. And here’s another example of a
prism, a triangular prism. The cross section, which is a
triangle, is the same all the way through the length of the prism. And here’s a circular prism. That circle shape is the same all
the way along the length of the prism. In fact, the circular prisms got
the special name of a cylinder.
Now, before we get too much into
volume, let’s think about a cube with each side of one unit. The cross-sectional area of that
cube would be one by one, which is one square unit. We can work out the volume of this
prism, this cuboid, by working out the cross-sectional area times the length. In this case, the length will be
the height of our cube. So our volume is one times one,
which is one. And because it’s volume, the units
will be cubic units.
So if we take our one unit cubed
cube and put it on top of another identical cube, we’ll have two cubic units. And a third makes three cubic
units, and a fourth makes four cubic units, and so on. But what would happen if we started
off with two of these cubes next to each other? Then, each time we added another
layer, we’d be adding on another two cubic units. So three layers gives us six cubic
units, and four layers is eight cubic units. So the general idea is that we take
our cross-sectional area and multiply it by the number of layers or the length or
height of our prism.
Now, as we said before, a cylinder
is just a prism with a circular cross-sectional area. So again to work out the volume, we
just work out the cross-sectional area and multiply it by the height. And so, the taller it gets, then
the greater the volume will be. To work out the area of a circle,
it’s 𝜋 times the radius squared, which you can write as 𝜋𝑟 squared, where 𝑟 is
our radius. And then, if we let the height of
our cylinder be ℎ, then since the volume is equal to the cross-sectional area times
the height, then we can write that the volume of a cylinder is equal to 𝜋𝑟 squared
ℎ. And we can use this result, this
formula, to help us answer questions involving the volume of a cylinder.
So let’s take a look at some
example questions.
Find the volume of the cylinder,
rounded to the nearest tenth.
So in this cylinder, we have a
radius 𝑟 of 4.2 feet and a height ℎ of 6.5 feet. We can use the approach that the
volume of the cylinder is equal to the cross-sectional area times the height. And since the cross-sectional area
is a circle, then this will be equal to 𝜋𝑟 squared, and the height we can write as
ℎ. Filling in the values of the radius
𝑟 and the height ℎ, we have 𝜋 times 4.2 squared times 6.5. We must be very careful that it’s
just the radius of 4.2 that is squared. So we have 𝜋 times 17.64 times
6.5. We can evaluate this using a
calculator as 360.2150137 and so on. And the units here will be cubic
feet since this is a volume.
Since we’re asked to round our
answer to the nearest tenth, then we check our second decimal digit and see if it’s
five or more. And since it isn’t, then our answer
stays as 360.2 cubic feet. And this is our final answer for
the volume of the cylinder.
In the next question, we’ll see an
example of finding the volume of a cylinder where we don’t need to use a
calculator. And we can do this by leaving our
answer in terms of 𝜋.
The volume of a cylinder is 𝑉
equals 𝜋𝑟 squared ℎ. Find the volume of a cylinder with
a radius of four centimeters and a height of 14 centimeters. Leave your answer in terms of
𝜋.
In this question, there are two
things to note. The first is that we haven’t been
given a diagram. And the second thing is we’re asked
to leave our answer in terms of 𝜋. So we won’t be simply typing our
values into a calculator and then rounding. We don’t need to draw a diagram,
but it can be helpful to help us organize our thoughts about a question. So on our cylinder, we have a
radius 𝑟 of four centimeters and a height ℎ of 14 centimeters. We can use the formula that the
volume of a cylinder is equal to 𝜋 times the radius squared times the height,
recalling that 𝜋 times the radius squared is the cross-sectional area, that’s the
area of a circle, multiplied by the height of the cylinder.
So for our 𝑉 equals 𝜋𝑟 squared
ℎ, we plug in the values of the radius and the height, which gives us 𝑉 equals 𝜋
times four squared times 14. And since four squared is equal to
16, we have 𝑉 equals 𝜋 times 16 times 14. Since 16 times 14 gives us 224, we
then have 𝑉 equals 224𝜋. Since this is a volume and both
units were given in centimeters, then our units will be cubic centimeters. We were asked to leave our answer
in terms of 𝜋. So our final answer then is 224𝜋
cubic centimeters.
We can often see more difficult
types of problems, which are written as word or story problems. Here, rather than explicitly saying
that there’s a cylinder and the radius and height are such and such, we have to work
out the meaning of the different variables from the context of the question. So let’s have a look at an example
of that.
Given that approximately 7.5
gallons of water can fill one cubic foot, about how many whole gallons of water
would be in this cylindrical water tank if it was full?
So if we imagine that our tank is
full of water, then we would find the volume of water in this tank by using the
formula that the volume of a cylinder is equal to 𝜋 times the radius squared times
the height. We aren’t given the radius 𝑟
here. But we are told that the diameter
is 20 feet. And since the radius is half of the
diameter, then we know that the radius would be 10 feet. So to find the volume, we take our
formula and plug in the values. So we have the volume is equal to
𝜋 times 10 squared times 12. And it’s important to note that
it’s just the 10 that’s squared and not the 𝜋 or the 12. Since 10 squared is 100, we then
have 𝜋 times 100 times 12, which is 1200𝜋 cubic feet.
For the moment, we can leave our
answer in terms of 𝜋 for maximum accuracy. If we start rounding to three
decimal places, we could carry these rounding errors through the calculation and the
final answer might be quite incorrect. We’ve worked out the volume of the
tank in cubic feet. But the question asks us how many
whole gallons of water would be in the cylindrical water tank. Each one cubic foot contains 7.5
gallons of water. So if there are 1200𝜋 cubic feet,
then there’s going to be 7.5 times as many gallons of water.
So the calculation we need to do is
to find the number of gallons, we multiply 1200𝜋 by 7.5. We can then put this into our
calculator to get 28274.3338 and so on gallons. But we will need to round this
since in the question we’re asked how many whole gallons of water and then checking
our first decimal digit to see if it’s five or more. As it is not, then our answer to
the nearest whole gallon is 28274 gallons.
Let’s now have a look at a question
where we compare the volume of a cylinder with the volume of another shape.
Which has the greater volume, a
cube whose edges are four centimeters long or a cylinder with a radius of three
centimeters and a height of eight centimeters?
So what we need to do here is
calculate the volume of the cube and calculate the volume of the cylinder and then
compare the two volumes. Starting with our cube, we can draw
a sketch where we have a cube of four centimeters by four centimeters by four
centimeters. And the volume can be found by
working out four times four times four, which is 64. And since our units are in
centimeters and it’s a volume, then our units will be cubic centimeters. We can then investigate our
cylinder, which has a radius of three centimeters and a height of eight
centimeters. The volume of the cylinder can be
found using the formula that it’s equal to 𝜋 times the radius squared times the
height.
So using the formula and plugging
in the values of the radius 𝑟 and the height ℎ, then we have that the volume equals
𝜋 times three squared times eight and being careful just to square the radius, the
three, and none of the other values. And since three squared is nine,
then we have 𝜋 times nine times eight. And nine times eight is 72, so we
have 𝜋 times 72. Using our calculator, we can write
this as 226.1946 and so on. And the units will be cubic
centimeters. And so, we’ve found the volumes of
both of these shapes. Since they’re both in the same
units, then we can easily compare our 64 cubic centimeters for the volume of our
cube and our 226.19 and so on cubic centimeters of our cylinder. So which one is greater? Well, our value 226.1946 is clearly
much larger than 64. And therefore, the cylinder has the
greater volume.
So now let’s summarize what we’ve
learned in this video.
We saw that a cylinder is a type of
prism with a circular cross section. To calculate the volume of a prism,
we find the area of the cross section and multiply it by the length, sometimes
called the height, of the prism. We can write this more specifically
as the formula the volume of a cylinder equals 𝜋 times the radius squared times the
height. And a top tip when we’re answering
a question, we must check were we given the diameter or the radius of the
cylinder. And finally, when answering story
problems, read the question carefully to find the relevant information and check
your units. And also, always consider drawing a
diagram, because this can be very helpful to help organize our thoughts on the
question.
