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Video: Volumes of Cylinders

Lauren McNaughten

Learn how to calculate the volume of a cylinder from its height and radius or diameter. Apply this knowledge in word problems and problems where working backward from knowing the volume to calculating a missing dimension is required.

10:56

Video Transcript

In this video, we are going to look at how to calculate the volume of a cylinder and then apply this method to a couple of related problems.

So let’s look at the method for calculating the volume of a cylinder. Now you’ve seen before how to calculate the volume of a prism, and there’s a reminder of this on the left-hand side of the screen. To calculate the volume of a prism, so any type of prism, we need to find first for the area of its cross section and then multiply it by the height or the depth of the prism.

So the cross section in this example here, which is a triangular prism, the cross section is that face on the front, the triangular face and I’ve shaded in green. And remember this method works for any type of prism at all. So a cylinder is just a special type of prism, and it’s a prism where the cross section is a circle.

So it’s this circular face that I’ve shaded in green here To calculate the volume of the cylinder then, I use exactly the same method. But when I’m finding the area of the cross section, I’ll be finding the area of a circle, so I need to recall the formula for calculating the area of a circle.

So the volume formula then: area of cross section, the area of a circle remember is given by 𝜋 multiplied by the radius squared, so 𝜋𝑟 squared, and then I have to multiply this by the height or depth of the prism. So in the example I have here that’s labelled as ℎ.

So this gives us our formula for calculating the volume of a cylinder specifically. The volume is equal to 𝜋𝑟 squared multiplied by ℎ, where 𝑟 is the radius of the circular cross section and ℎ is the height of the cylinder.

So let’s look at applying this formula to an example. The question asked us to calculate the volume of the cylinder shown, and we have the diagram here. Now first let’s recall that volume formula that we need. And remember, it’s this: volume is equal to 𝜋𝑟 squared ℎ. So it’s just a question of substituting the correct values for this question into that formula now there is one detail we need to be careful with. The formula uses 𝑟, which remember represents the radius, whereas in the question if you look at the diagram carefully, we’re not actually given the radius, we’re given the diameter of the circle.

That’s not a problem because of course the radius is just half of the diameter, but it is something you need to be aware of when working with circles: have you been given the radius or have you been given the diameter? And which one do you need?

So let’s go ahead and substitute the relevant pieces of information into our formula for calculating the volume of this cylinder. So the volume is equal to 𝜋 multiplied by 𝑟 squared. Well if the diameter is eight centimetres, then the radius of the circle is four centimetres, so four squared. And then the height of this cylinder is ten centimetres. So my calculation for the volume is 𝜋 multiplied by four squared multiplied by ten.

If I evaluate that it gives me an answer of one hundred and sixty 𝜋. And sometimes you may be asked to leave your answers like that because they’re exact values while there are values that have to be rounded or perhaps you might be working without a calculator, in which case you wouldn’t be able to proceed to the next stage of the calculation. However, I have got a calculator so I’m gonna evaluate what a hundred and sixty 𝜋 is equal to.

And it’s equal to five hundred and two point seven centimetres cubed. That answer has been rounded to one decimal place or the nearest tenth. Just a note about units there, we’re talking about volume so it’s cubic units: centimetres cubed or metres cubed or basically whatever the units are in the question, cubed.

Okay let’s look at our next question: a cylindrical container has a diameter of seventy centimetres. Water is poured into the container until it reaches a height of one point five metres. What, in litres, is the volume of water in the container? So I think for this question, it’s sensible just to sketch a quick diagram so we can visualise the situation. So we’re gonna have a diagram of a cylinder, and we’ll put the relevant information onto it.

So here is my cylindrical container. I’m told that the diameter of this container is seventy centimetres, so that’s this measurement here, and I’m told that the water is poured in until it reaches a height of one and a half metres, so something like this. Now the question asked us to calculate the volume of this water in litres, so at a later stage we’ll have to think about converting between centimetres cubed or metres cubed and litres.

For now, there’s one other detail within the question that I need to be aware of, and it’s the fact that my measurements have been given in different units: this diameter is in centimetres, whereas the height of the water is in metres. And I need them both to be the same before I do my calculation.

So I’m going to convert this one point five metres into a hundred and fifty centimetres. So let’s calculate the volume of this water. Well the water in the container is a cylindrical shape, so I can use my volume formula of 𝜋𝑟 squared ℎ in order to calculate it.

And as before, I just have to be a little bit careful because I’ve been given the diameter rather than the radius of the cylinder again. But if the diameter is seventy centimetres, then the radius is gonna be half of that so it’s going to be thirty-five centimetres. So my calculation then, 𝜋 multiplied by thirty-five squared multiplied by a hundred and fifty.

If I evaluate that, it gives me one hundred and eighty-three thousand seven hundred and fifty 𝜋. Again I could leave my answer at this stage to keep it exact, but I’m gonna use my calculator to work out what that is as a decimal. And so this is five hundred and seventy-seven thousand two hundred and sixty-seven point seven centimetres cubed.

Now remember the question asked us to calculate this volume in litres, so I haven’t finished my working out yet. I need to think about how I convert from centimetres cubed into litres And so you need to recall a key piece of information for this, which is that one centimetre cubed of space can contain one millilitre of liquid. So it’s a one-for-one conversion between centimetres cubed and millilitres.

What this means then is this answer that I have in terms of centimetres cubed, well I can convert it directly into millilitres. So it’s just exactly the same number of millilitres as it was centimetres cubed. And then if I want to convert my answer to litres, I need to divide it by one thousand because there are of course one thousand millilitres in one litre.

So I’ve divided it by a thousand and then I’ve chosen to round my answer to one decimal place, so five hundred and seventy-seven point three litres is the volume of water in this container.

So a couple of things just to remind you of from this question: we had to be careful with the fact that we had differing measurements, centimetres and metres, for the dimensions of the cylinder, and we had to convert them to be the same before we could do any working out; we also needed to remember the conversion between centimetres cubed and millilitres in order to give our answer in the format that it was asked for in the question.

Right, the final question that we’re going to look at in this video. We’re given a cylinder and we’re told that its volume is one thousand five hundred and eighty-three point four centimetres cubed. If we look at the diagram, we’re given the height of the cylinder; it’s fourteen centimetres. But we’re not told what the diameter is, and so that’s our objective: to calculate the diameter of this cylinder.

So you may want to pause the video for a moment at this point and just think about how you’re going to use the information you’ve been given in order to calculate that diameter. I’m going to recall first of all that volume formula that we’ve been using throughout this video.

And of course it’s this: the volume is equal to 𝜋 times 𝑟 squared times ℎ. Now as we know, we haven’t been given the diameter so therefore we don’t know the radius. But we do know the volume; we know it’s equal to one thousand five hundred and eighty-three point four centimetres cubed. So what I’m able to do is use the information I’ve been given to set up an equation, using the volume and the height, and then I’ll be able to solve that equation in order to work out the value of 𝑟.

So let’s start formulating this equation. The volume, so that is one thousand five hundred and eighty-three point four. Well it’s equal to 𝜋 multiplied by 𝑟 squared — I don’t know what that is, so I’ll keep it as 𝑟 squared — multiplied by ℎ, which is fourteen.

So there’s the start of my equation. Now if I just simplify the right-hand side of this equation a little bit so I can just write it without the multiplication signs, cause we don’t need those in the algebra here.

Now I want to solve this equation in order to work out the value of 𝑟. So I have fourteen 𝜋 multiplied by 𝑟 squared, therefore my first step is going to be to divide both sides of this equation by fourteen 𝜋. And so that gives me this stage of working out here, that 𝑟 squared is equal to one thousand five hundred and eighty-three point four over fourteen 𝜋.

Now if I evaluate that — and I’ll just write the two sides of the equation the other way round so that 𝑟 squared is on the left — then it gives me that 𝑟 squared is equal thirty-six point zero zero zero eight four eight, and so on. The final step in calculating 𝑟 is I need to take the square root of both sides of this equation.

So this gives me 𝑟 is equal to six point zero zero zero zero seven zero six, and so on. So to any sort of reasonable degree of accuracy, that’s 𝑟 is equal to six. So the radius of this cylinder then is equal to six centimetres. Remember, I’m not asked to calculate the radius; I’m asked to calculate the diameter. So the final step I just need to double this value in order to find the diameter of the cylinder.

So my final answer then is that the diameter of this cylinder is twelve centimetres. This question then is an example of working backwards from knowing the volume to calculating one of the measurements. You could perhaps be asked a similar question where you did know the radius or diameter but it was the height of the cylinder that you were looking to work out, and of course you’d proceed in a very similar way.

So to summarise then, we’ve looked at the formula and the method for calculating the volume of a cylinder; we’ve applied it to a more straightforward problem, where it’s just a case of substituting the relevant measurements; we’ve applied it to a worded problem; and then finally we’ve looked at a problem that involves working backwards from knowing the volume to working out one of the dimensions of the cylinder.