Lesson Video: Finding the Volume of a Cylinder Mathematics • 8th Grade

In this video, we will learn how to calculate volumes of cylinders and solve problems including real-life situations.

12:35

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.

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