Video Transcript
In this video, we’re gonna be
talking about how to find the volume of a cylinder. First, we’re gonna take a look at
prisms and how to work out the volume of a prism. And then, we’re gonna explain how a
cylinder is a circular prism. Finally, we’ll be looking 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 blue stripy bit here. And if I was to cut this prism at
any point, this cuboid, at any point along here, so let’s say through here like
this, and look at the slice that I get, I would still have exactly this same cross
section.
Here’s another example of a prism,
a star-shaped prism. That cross section, which is a star
shape, is the same all the way through the length of the prism. And here’s a circular prism. That circular shape is the same all
the way through the length of the prism. In fact, that circular prism’s got
the special name of a cylinder.
Now, before we go too far thinking
about volumes, let’s talk about a cube with each side of length one unit. The cross-sectional area of that
cube will be one unit by one unit, which is one unit squared. Now, we can work out the volume by
multiplying the cross-sectional area by the length, or in this case the height of
the prism. So, that’ll be one times one, which
is equal to one. And because it’s volume, it’s units
cubed.
Now, if we take our one-unit-cubed
cube and pile it on top of another identical one, we’ll have two cubic units. Now, a third makes three cubic
units. And a fourth makes four cubic
units, and so on. But what if we started off with two
of these cubic units next to each other? Now, each time we add an extra
layer, we’re adding another two cubic units. So, three layers gives us six cubic
units. And four layers gives us eight
cubic units. So, the general idea is that for
volume you’re taking the cross-sectional area that we had here and multiplying it by
the number of layers, or the length, or the height of that 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. The taller it gets, the greater the
volume. Now, remember, to work out the area
of a circle, it’s 𝜋 times the square of the radius. So, if we call our radius 𝑟, the
area is equal to 𝜋 times 𝑟 squared. And if I let the height of my
cylinder, or the length of my cylinder, be ℎ, because the volume is equal to the
cross-sectional area times the height, we can say that the volume is 𝜋𝑟 squared
times ℎ. And that’s the result that we’re
gonna be using in our examples in the rest of this video.
For example, find the volume of
the cylinder rounded to the nearest tenth. And the circle on the end of
our cylinder has a radius of 4.2 feet. And the cylinder’s got a height
of 6.5 feet.
So, we’ll mark up 𝑟, the
radius, is equal to 4.2 and ℎ, the height, is equal to 6.5. So, our approach is gonna be
that the volume is equal to the cross-sectional area times the height. And since the cross-sectional
area is a circle, the area is gonna be 𝜋 times the radius squared. So, that’s 𝜋 times 4.2
squared. Now, it’s important to remember
that it’s only the 4.2 that is squared, not the 𝜋. So, that’s gonna be 𝜋 times
17.64, which gives us an area of 55.41769441 and so on square feet.
But to work out the volume,
remember, we do need to multiply by the height as well. So, let’s add that to our
working out. And 55.41769441 times 6.5 gives
us 360.2150137, so on, so on, so on cubic feet. But the question asked us to
round our answer to the nearest tenth. So, I’m gonna cover everything
up after the tenth and then just do a sneaky peek at the next digit to see
whether I need to round that to up or not. Well, the next digit is only a
one. And if it was five or above,
then we’d be rounding the two up to a three. But it’s not; it’s only a
one. So, we’re gonna keep it as a
two. So, our answer to the nearest
tenth is 360.2 cubic feet.
Now, let’s look at a similar
example. But this time we’ve been given
the diameter of the cylinder rather than the radius.
Now, remember, the radius is
half of the diameter. So, to work out the radius, we
just need to divide 14 by two, or multiply it by a half. And that gives us seven
inches. And the formula for our volume
is 𝑉 equals 𝜋𝑟 squared ℎ. And so, substituting in the
numbers for the radius of seven inches and the height of 13 inches, we’ve got 𝜋
times seven squared times 13. Again, it’s important to
remember that it’s only the seven that’s squared and not the 𝜋. So, that’s 𝜋 times 49 times
13. And when we put that into our
calculator and round to the nearest tenth, we get 2001.2 cubic inches.
In this example, we’ve been
asked to find the volume of a cylinder with a radius of four centimetres and a
height of 14 centimetres. We’re also told that we’ve got
to leave the answer in terms of 𝜋.
Now, there are a couple of
things here. One, we haven’t been given a
diagram. And two, we’ve got to leave our
answer in terms of 𝜋, so this isn’t a matter of just punching the number into a
calculator and doing any rounding. Now, you don’t need a diagram,
but very often drawing a diagram helps you to organize your thoughts about a
question. So, I would recommend actually
doing a quick sketch. So, there’s our cylinder. It’s got a height of 14
centimetres and a radius of four centimetres.
Next, we can write out the
formula for the volume. The volume of a cylinder is 𝜋
times the radius squared times its height. And we can substitute in the
numbers we’ve been given, so 𝜋 is equal to four squared times 14, which is 𝜋
times 16 times 14. And 16 times 14 is 224. So, our answer is 224 times
𝜋. Now, from the question, both of
our measurements were given in centimetres. So, the volume is gonna be in
cubic centimetres. So, there we have it. That’s our answer. 224 𝜋 cubic centimetres. So, when the question says
leave your answer in terms of 𝜋, it means express it as a multiple of 𝜋.
Now, we can make things a
little bit more difficult by turning these things into word or story
problems. So, rather than just explicitly
saying that we’ve got a cylinder and telling you what the radius and the height
are and just doing that calculation, you have to work out the meaning of the
different variables from the context of the question.
So, let’s have a look at some
examples like that.
Given that approximately 7.5
gallons of water can fill one cubic foot, about how many whole gallons of water
would be in a cylindrical water tank with diameter 20 feet and height 12 feet,
if it was full?
Okay, first, let’s do a little
diagram. Here, we have our cylindrical
tank completely full of water, depth, or height, of 12 feet and diameter of 20
feet. So, first, we can write down
that the volume is equal to 𝜋 times the radius squared times the height. Now, we can plug in the numbers
that we know. Well, the radius is half of the
diameter, so half of 20 is 10. So, the radius squared is going
to be 10 squared. And it’s important to remember
that it’s just the 10 that’s squared, not the 𝜋 as well. And the height is 12, so we’ve
got to multiply that answer by 12.
So, this calculation is 𝜋
times 10 squared, which is 100, times 12, so 𝜋 times 1200, or 1200𝜋 cubic
feet. Now, for the moment, I’m gonna
leave my answer in terms of 𝜋 for maximum accuracy. If I started rounding to a few
decimal places, I would carry these rounding errors through my calculation and
my final answer might be quite incorrect. Now, we’ve worked out the
volume of the tank in cubic feet, but the question says how many whole gallons
of water would be in the cylindrical water tank.
Now, each one cubic foot
contains 7.5 gallons of water. So, if there are 1200𝜋 cubic
feet, there are gonna be 7.5 times as many gallons of water. So, the calculation we need to
do to work out the number of gallons is 7.5 times 1200𝜋, which I can do on my
calculator. Now, it’s okay to round right
at the end of the question. And the question said about how
many whole gallons, so I need to round to the nearest whole gallon. So looking at our number here,
that’s gonna be 28274. So, we can write our answer out
nice and neatly at the end, 28274 gallons of water.
Which has the greater volume, a
cube whose edges are four centimetres long or a cylinder with a radius of three
centimetres and a height of eight centimetres?
So, what we’ve got to do here
is calculate the volume of the cube and also calculate the volume of the
cylinder and then compare the two. So, first, the cube, let’s draw
a sketch, four centimetres by four centimetres by four centimetres. And the volume is just gonna be
four times four times four. And since the length units were
centimetres, our volume is gonna be in cubic centimetres. And four times four times four
is 64. So, the volume of the cube is
64 cubic centimetres.
Now, a quick sketch of the
cylinder. And use the formula the volume
is equal to 𝜋 times the square of the radius times the height. Now, let’s remember that that
squared only applies to the three. It doesn’t apply to the 𝜋. So, we’ve got 𝜋 times three
squared times eight. And three squared is nine. So, nine times eight is 72. So, we’ve got 𝜋 times 72. Now, it doesn’t ask for a level
of accuracy in the question. But I’ve rounded that to two
decimal places to give me 226.19 cubic centimetres.
So, again, the measurements
were in centimetres, the volume is in cubic centimetres, and the two numbers
that we’ve got to compare are both in the same units, cubic centimetres. Now, we can compare those. And 226.19 is clearly a lot
larger than 64, so the cylinder has got the greater volume.
Now, let’s summarize what we’ve
learned. First, a cylinder is a type of
prism with a circular cross section. Next, to calculate the volume of a
prism, you find the area of the cross section and multiply that by the length, or
sometimes called the height, of the prism. The volume of a cylinder, 𝑉, is
equal to 𝜋 times the square of the radius times the height.
And a top tip, always check were
you given the diameter or the radius of the cylinder in the question. That’s really important. And finally, when answering story
problems, make sure to read the question carefully to find the relevant information
and check your units. Also, always consider drawing a
diagram cause it can be really helpful to organize your thoughts on the problem.
