Lesson video

Hello, I'm Mrs. Lashley and I'm really looking forward to working with you as we go through this lesson today.

So hopefully by the end of this lesson we will be able to work out the volume of any cylinder by using the constant cross-sectional area to determine the volume.

On the screen are some words that we'll be using during the lesson that you have met before in your learning.

But you may want to pause the video now whilst you just re-familiarize yourself before we move on.

So the lesson's got two parts to it.

The first part, we are gonna work with the volume of a cylinder and be able to calculate the volume for any cylinder, whereas in the second part we're going to make use of the volume of the cylinder in other questions.

So let's make a start on working out the volume of any cylinder.

We're gonna be looking at the volume of cylinders today, but previously you may have learned about volumes of prisms. So what's the difference between a cylinder and a prism? So firstly, what is the same? Well, they're both 3D shapes.

They're both three-dimensional.

And secondly, they have a uniform cross-section that runs through the whole length of the shape.

So why is a cylinder not a prism? What makes them different? Well, what makes them different is that the cross-section for a cylinder is a circle, and that is not a polygon.

And the definition for a prism is that that cross-sectional face is a polygon.

So because a circle is not a polygon, then a cylinder is not a prism.

Izzy said, "It's not a prism, but they do have a similar structure." And that goes back to this idea that there is this uniform cross-sectional face that runs throughout the length of the prism or the cylinder.

So both of these 3D shapes have got a uniform cross-section, and that is where they are connected in terms of their structure.

So a quick check.

A cylinder is a prism, true or false? Okay, so I'm hoping you've remembered that it is false, it's not a prism.

So now justify your answer.

Your justification should have been A, you need to know that a prism has a polygon as its cross-section.

The circle is not a polygon and therefore a cylinder is not a prism.

Izzy and Aisha are discussing how to get the volume of a cylinder if the structure is similar to a prism.

And so Izzy's saying, "We can work out the volume of a cylinder in a very similar way to a prism." And Aisha is agreeing with her by saying, "Yes, because it's like a stack of circle slices." So volume, when we've looked at volume in the past, you would've thought about it being stacks or slices of the same shape because that was the uniform cross section.

And that's not gonna be different with a cylinder.

A cylinder is not a prism, but it does have a uniform cross section, it just happens to be a circle.

So it's like a stack of circle slices.

So if this was our first slice, we can have more and more of these slices on top of each other and we can see a cylinder growing.

And the heights would be that depth between the top face and the bottom face.

Those two faces are identical.

They are congruent to each other because of the uniform nature.

So a reminder probably about what volume actually is as a concept.

So volume is the space inside of a 3D solid or how much space a 3D solid takes up.

And it is calculated by multiplying three perpendicular lengths, and the formula for a prism, so on the screen here is a keyboard, which is a prism.

Remembering a cylinder is not a prism, but going back to prisms for the moment, it's the area of the cross section and the area of any 2D shape has been created by multiplying two lengths together.

And then we multiply it by the length of the prism, because of this idea of sort of stacks or slices and layers.

And so here you can see that we've got length multiplied by length, multiplied by length, and we could say that has been cubed.

So the units of volume can be cubic centimetres, cubic millimetres or cubic metres.

Can you think of any others? Pause the video whilst you might be thinking about that.

And then when you're ready to check press play.

Some other ones that you may have thought about is a cubic foot or a cubic inch, and they are imperial measures.

Cubic yard would've worked as well.

So volume is about how much space is inside of a 3D solid or how much space it takes up.

And we have cubic units because of this idea that three perpendicular lengths have been multiplied together.

Heading back to cylinders then.

So we know that cylinders have got a similar structure to a prism.

The difference is the shape of the base.

So a cylinder and any cylinder has got a circular cross section.

It's always a circle if it is a cylinder.

And the area of a circle is pi r squared, you'd have covered that in prior learning.

And here you can see that it's r centimetres times by r centimetres, r squared, and they are your two perpendicular lengths for the area for this circle.

Pi is that irrational number and is a relationship between the circumference and the diameter of all circles.

And so the volume of a cylinder where we need to go from having an area to a three-dimensional measure is multiplying it by the height.

So every cylinder has got a circular face, and so it's pi r squared for the circle.

And then to get the volume of the cylinder, we multiply it by the height.

So the volume of any cylinder is, you can see there on the screen, pi r squared multiplied by height.

And that can be, we can change the height to an h where h is the height of the cylinder.

And so pi r squared h.

Lucas is reminding us though, "But because of order of operations, you must square the radius before anything else." And here is a worked example on the screen for this particular cylinder.

It's got a radius of four centimetres and a height of nine.

So into the formula, pi times four squared, multiplied by nine.

What Lucas told us to do, square first.

So the four squared has been evaluated to 16, then we can evaluate 16 multiplied by 9, 144 pi cubic centimetres.

Leaving it in terms of pi is the most accurate, we've got no rounding error there, but if you are asked to give it to a specific degree of accuracy, then you can round it, but you do need to state the degree of accuracy.

So here we've got 452 to three significant figures, SF meaning significant figures.

So on this check, I would like you to identify the errors in this solution.

So it is a calculation for the volume of the cylinder that you can see.

Pause the video to figure out where has the calculation gone wrong, what errors have made? When you press play, we'll go through the answers.

So first up, the only thing that should be squared is the radius, but the whole product of 5 and 12 is being squared here.

So only the radius should be squared.

So that's an error.

So therefore a carry on error is that the answer is now wrong because of the calculation being wrong.

And also did you notice that the units were not right? If it's a volume, then the units should be a cubed unit.

All of the units of length given on the diagram were centimetres so the volume would've been cubic centimetres.

Here it says square centimetres.

Here we've got a question where Sophia has been into a shop and has noticed some packaging that looks like this shape.

So we've got a sort of 3D solid version of the shape, and then we've got a plan view, so that's looking from the top, and a side view.

So she describes it as a cylinder with a narrower cylinder removed.

And she wants to know is there a way to calculate the volume of this packaging.

So this packaging that she's got, her description is a cylinder with a narrower cylinder removed.

Alex says, "We'll calculate the area of the cross section, then multiply it by the height." And that has been the volume for a prism, calculate area of the cross-section multiplied by the height.

We know that cylinders have a similar structure to a prism.

So I think Alex is going for, well, surely that's gonna be the way that this works.

Area of the cross-section, multiplied by the height.

Sam suggests that we should subtract the volume of the centre cylinder from the volume of the larger cylinder.

So sort of to think of it as two separate cylinders and then subtract it.

And Sophia did describe the structure as it being removed and subtraction would be doing the removal.

Both Alex and Sam have tried to calculate the volume here of that packaging now that we know their dimensions.

So we've got 26 centimetres as the diameter from external edge to external edge.

We've got eight centimetres as the internal radius and six centimetres as the height of the packaging.

We can see that again on the plan view and the side view.

And hopefully you can see on the screen that actually Alex and Sam get the same volume.

They get the same answer.

So Alex went for cross-sectional area and his subtraction happened there.

The area, he's looking at the plan view and seeing this as a circle with a smaller circle removed.

And so he's worked out the area of the larger circle, which would have a radius of 13, and the smaller circle has got radius of eight.

And the subtraction, the removal happened at that point of his calculation, and then he multiplied it by six because that was the height of the cylinder or the shape that was cylinder-like, 630 pi.

Whereas Sam went for the large cylinder, subtracting the inner cylinder and still gets 630 pi.

If you wrote those two calculations out as one calculation, you would see that they are equivalent.

And basically Sam has multiplied by six on both and then subtracted, whereas for Alex, subtracted and then multiplied by six, and Alex's area would've been in brackets.

And if we had multiplied out the brackets, like expanding brackets, we'd get the exact same calculation and that's why we have the same volume.

So they're both correct, they just both saw it in slightly different ways.

Here is a check with a different shape, but similar to what Sophia had seen in the shop.

So a necklace pendant is cylindrical, its cross-section is shown here.

It's three millimetres deep.

Which of the calculations shown would work out the volume of the pendant? So pause the video whilst you're working through those, there might be more than one.

And then when you're ready to check, press play.

So part C was the correct calculation or a version of the correct calculation for the volume.

So just very briefly to go through why A, B, and D were not correct.

Both A and B have used 32 and 4 from the diagram, but 32 and the 4 were diameters, the radius needs to be half of that.

An area of a circle is pi r squared, pi times radius squared.

So they needed to convert the diameter into its radius.

And for D, they did that correctly for 16, the 32 has been halved, but they didn't do it on the 4.

So it's been managed on one part, but not on the other.

Whereas C, and they've gone for sort of Alex's method here that they've run the area of the cross section and then multiplied it by the depth of the pendant.

But they have remembered to convert the diameters into radius before they start the calculation.

So you're now gonna do some tasks about volume of a cylinder.

Here is question one and this question is the volume of a cylinder is pi r squared h, where r is the radius, and h is the height, which of the following are equivalent? So you've got a whole host of algebraic equivalent formulas written there, but which ones are equivalent to pi r squared h.

Pause the video whilst you're figuring those out.

And then when you're ready for question two, press play and we'll move on.

So here is question two.

You need to calculate the volume of the following cylinders and do leave your answers in terms of pi.

Think about the units depending on each question.

That's for A, B and C.

There is a diagram to support you.

For D and E, there isn't a diagram, so you might wanna sketch yourself a diagram if that's more helpful.

But read it carefully and calculate the volume.

So there are five cylinders, you need to work out the volume of each one.

Pause the video whilst you're doing that and then when you come back, we'll go onto question three before we then go onto the answers.

Okay, so here is the last question of this task.

So a metal washer factory forms a long metal pipe before slicing them into individual washers.

What volume of metal is used to create the washers? So here we've got our diagram of this metal pipe, and then they will slice it into the individuals.

So what volume of metal is used to create the washers? Pause the video whilst you're working out that.

And then when you come back, we're gonna go through all of the answers to this task.

Question one, there were four equivalent formulas to pi r squared h.

So B, it was just the order of the multiplication that had changed.

The square still belongs to the r, on B.

On E, it's now just been written out as, it should have been written out as a product with the multiplication symbols, and r squared has been written as r multiplied by r, which by definition that's what that means.

And F is a version of that as well.

Again, just the reordering of the multiplication.

And because multiplication is commutative, then that's absolutely fine.

And finally, H, again, it's a reorder of the original formula where the squared is still part of the r.

Question two, you had five volumes to work out.

Three of them, you had a diagram to support you, and two of them were just worded.

You needed to leave all of them in terms of pi, regardless of what units.

So A, you had the radius and you had the height.

So it was just a case of substituting it in and evaluating.

So 63 pi cubic centimetres.

B, the 2 was the diameter.

So you did need to recognise that and say that the radius is 1 before using the formula.

So 4 pi cubic metres.

C, we had mixed units, you've got millimetres and you've got centimetres.

It's up to you which way you converted.

You could have put them in millimetres or you could have put them in centimetres.

Both of the answers are on the screen depending on which way you went.

And both of them need to be in terms of pi.

Part D, a cylinder with a radius of 8 and a depth of 12.

So again, just being comfortable that height, length, and depth are all meaning the same thing when it comes to the formula.

It just depends on how we sort of view the cylinder.

So D, 768 pi cubic millimetres.

And finally E, a diameter and a height.

So you did need to change the diameter of the radius before you used the formula and 69.

12 pi cubic centimetres.

Finally we had question three, and again, depending on which unit you went for.

So if you did it in millimetres, they were all in millimetres, that's probably the one you've used because they were all in millimetres.

You'd have got 305,645.

5 cubic millimetres, that's to one decimal place, or 97,290 pi.

And if you converted them to centimetres, then you'd have 97.

29 pi cubic centimetres or 305.

6 to one decimal place cubic centimetres.

Right, so we're now gonna move on to the second learning cycle, which is where we're gonna make use of the volume of the cylinder.

So we've been able to calculate the volume of any cylinder now, now we're gonna make use of it.

The formula for the volume of any cylinder includes three variables.

The volume itself, the radius, and the height.

Andeep has recognised that if you have two of the three variables, then you can work out the third.

So let's look at this problem on the screen.

We know the volume of the cylinder, we know the radius of the cylinder, what we don't know is the height, but because we have two of them, with two of the variables, we've got the volume and we've got the radius, then we should be able to calculate the height.

So by substituting what we have into the formula for the volume, we can then start to solve this as an equation.

So here we've got 1,215 pi equals pi times nine squared.

So that's the area of the cross-sectional face.

And then multiplying it by height, which is what we don't know.

If we evaluate the area of the circle, we get 81 pi and that is multiplied by the height, which is equal, is equivalent to 1,215 pi.

By using inverse operations because it was being multiplied, so we can use division to help us solve here, then the height would be 15 centimetres.

So because we had two of the three variables included in the formula, we can work out the third.

So I'm gonna run through an example and then for your check, you'll do a similar question by yourself.

So work out the height of a cylinder, which has a volume of 225 pi and a diameter of 10.

So note that we have a diameter.

A bit of preparation before we start is that we have to convert that diameter to the radius of being five.

If a diameter is 10, we know that the radius is actually five.

Now we can substitute what we have into this formula.

This is the formula for any cylinder, which is a really useful thing because all cylinders have got that circular base.

So we know the volume, we know the radius is five by substituting it in, then we evaluate, we're now at a point where we can solve to work out that missing dimension, that missing length, which is h in this case.

And that would be nine.

So the height of a cylinder, which has a radius of five and a volume of 225 pi would be nine.

So this is your question for you to solve.

So pause the video whilst you're doing that, and then when you're ready to check it, press play.

Again, the diameter was given.

So your first sort of preparation point is to change that to a radius by knowing that a diameter is twice the length of a radius, or conversely, you could think of that as the radius is half the diameter.

Then we've got that formula where we're gonna substitute our given information in, the two of the three variables, which allows us to evaluate it and then solve it.

So the answer to your question was that the height would be seven centimetres.

So Andeep has already mentioned that if we have two of the three variables in the volume of a cylinder question, then you can always calculate the third.

So here our problem is that we have the volume, we have the height or the length of the cylinder.

What we don't know is the area of the circle or the base, but if we break down what the actual formula is, the actual formula is area of a circle multiplied by the height.

So we should be able to work out the area.

We can think of that as its own variable because that is dependent on the radius.

So substituting what we do know, we can say that pi r squared, which is the area of a circle, is 64 pi.

So even if it's not the radius that we're trying to calculate, we can get the area because the area is dependent on the radius.

But Andeep goes further to say, "But if you know the area of the circle, then what other information can you work out?" Well, if you know the area, you can get the radius because the area is dependent on the radius.

Continuing with that example, we've got down to that pi r squared, or the area of the circle, is 64 pi.

So if we divide by pi, then we know that radius squared is 64 and we can square root.

We know that 64 is a square number, you probably know that that means it's eight squared, so the radius would be eight centimetres.

So if we had the volume and we had the height of the cylinder, we can work out the area of the cross section, but we can go further and we can work out the radius of the cross section or the cylinder.

And actually if you know the radius or you can get to the radius, then you also know the diameter.

So still working with this example, if we now know that the radius of this cylinder is eight centimetres, then that infers the diameter is 16 centimetres because of the relationship between radius and diameter.

So here is a check for you.

So given that the radius of the cylinder is six centimetres, which of the following are true? Pause the video whilst you think about them individually.

And then when you're ready to check, press play.

Okay, so firstly B is correct.

If the radius is six, then the diameter is 12.

Also the area of the cross section would be 36 pi.

And that's because if the radius is six, pi r squared, pi times 6 squared is equivalent to 36 pi.

We have no information about the height of the cylinder, so it could be 12, it might be 12, but we don't know that for certain.

The radius does not give you any information about the height.

And because we don't have any information about the height, we have no idea what the height is, we couldn't tell you what the volume is as a numerical value.

You'd be able to give the volume as 36 pi h where you still need to substitute the value of the h, but we can't say that it's 36 pi.

So B and D were the correct answers to your check.

Here we've got Laura, who is buying a candle and a gift box for her grandma.

The candle is cylindrical, it has a volume of 256 pi and is 16 centimetres tall.

The gift box is 75 millimetres by 75 millimetres by 165 millimetres.

So it's a box, a cuboid, with those dimensions.

Will the candle fit? So if she buys that candle and that gift box, will she be able to box it up for her grandma? Alex tells her this doesn't fit.

"It doesn't fit." So let's have a look at why or how Alex got to that conclusion.

So here's a quick sketch of the box.

So 7.

5 centimetres is equivalent to 75 millimetres, 7.

5 centimetres, and then 16.

5 centimetres.

So it's a square based cuboid.

The candle was 16 centimetres tall and the box is 16.

5 centimetres tall.

So the candle is shorter than the box, so it would fit by height.

The volume of the cylinder.

So remember the candle was a cylindrical shape.

So the volume of the cylinder is 804.

2 cubic centimetres to one decimal place.

And the box is 928.

1 cubic centimetres to one decimal place.

So it has less volume.

So the box takes up more space than the candle, and the candle is shorter than the box.

So why doesn't it fit? I just want you to think about what else might be affecting that the candle does not fit inside this box, even though the box is tall enough and by volume has got enough space.

So the circular face of the cylindrical candle needs to fit within the square face of the box.

So we've already recognised that it's not to do with the height.

The height of the candle is not the issue.

So the issue is the cross section.

The circular face on the candle needs to fit within the square face of the box.

And so we need to figure out what the size is.

So if we know the volume of the candle and we know the height of the candle, we can go through all of those stages that we've seen previously, substituting it in, and solving it to work out that the radius of this candle is four and therefore the diameter of the candle, so the longest part or the widest part of the candle would be eight centimetres.

And that's the reason it doesn't fit in the box because the box is only 7.

5 by 7.

5.

And so the candle will not fit in the box because it is wider than the box is.

So despite the fact it was tall enough, it had enough space within it, the candle itself would not fit.

So here is a check similar to that.

A box is being made for a cylindrical sculpture.

The cylinder has a volume of 11,250 pi and a height of 50 centimetres.

What is the smallest the box can be so the sculpture fits? So this time you are trying to come up with the dimensions to ensure that the sculpture fits.

Pause the video whilst you work that one out and then press play and we'll go through it.

Okay, so first up, let's use the information we know.

We know the volume is 11,250 pi.

We know that the height of the sculpture is 50 centimetres.

What we don't know yet is the radius or the diameter, the circular face.

So substituting it into the formula for the volume of the cylinder.

We can go through the stages, dividing both sides by 50 pi gives us the radius squared is 225, which means that the radius is 15.

So this sculpture has got a radius of 15 or a diameter of 30 centimetres.

So we need to make sure the box is large enough to fit this cylindrical shape.

, this cylindrical sculpture that has a maximum width of 30 centimetres, that's the longest length across the circle, and a height of 50 centimetres.

So the box will need to be at least 30 centimetres by 30 centimetres.

So surround it in a square, and then by 50 centimetres in height in order to fit the sculpture.

Well done if you got that one correct.

So now you are onto your practise part of the lesson for this second learning cycle.

So all of these cylinders have the same volume and you need to calculate the missing lengths on the ones that have the missing length.

Pause the video whilst you're working out all of those missing lengths.

And then when you're ready, we'll go through the answers.

Okay, so all the answers are on the screen right now.

The way to get into this task was that there was one of the cylinders that you had enough information to work the volume out.

Because you were told they all had the same volume, you needed to work that volume out, and that was 360 pi.

And then you could go through all of the stages of working out diameters, radius and lengths of the circles.

So A was three centimetres, so that was the radius.

For B, it was 0.

9 centimetres.

For C, it was 22.

5 centimetres.

For D, it was two centimetres, and for E it was 90 centimetres.

So in summary, today's lesson has been about working out the volume of any cylinder, and because cylinders have a similar structure to prisms, this was the way we went into the question.

The difference between the cylinder and the prism was the cross-section, that a prism needed to have a polygon as its cross-section, and a cylinder doesn't have that.

But otherwise the structure is very similar.

So the formula to find the volume of a cylinder is pi r squared, which is the area of the circle, the cross-section, and then multiply it by h, which is the height.

And you can use the formula to work out the radius, the diameter, or the height if you know the volume and one of the others.

So out of the three variables in the formula, if you knew two of them, you could work out the third.

Really well done today and I look forward to working with you again in the future.