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Lesson Video: Volumes of Cylinders Mathematics • 8th Grade

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

11:11

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.

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