Lesson video

Hi, and thank you for joining me.

My name is Dr.

Roleson, and I'll be guiding you through this lesson.

So let's get started.

Welcome to today's lesson from the unit of 2D, and 3D shape with surface area and volume, including pyramids, spheres, and cones.

This lesson is called the volume of a cone, and by the end of today's lesson we'll be able to calculate the volume of a cone.

Here are some previous keywords that will be useful during today's lesson.

So you might want to pause the video if you want to remind yourself what any of these words mean.

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

This lesson is broken into two parts.

In the first part, we're going to learn how to find the volume of a cone, and then in the second part, we're going to use the volume of a cone to work out other things.

But let's start off with finding the volume of a cone.

Here we have a cylindrical container with a height of 21 centimetres.

But hang on.

This lesson is about finding the volumes of cones.

So why we're looking at a cylinder? Well, the volume of a cone can be calculated using the displacement of water from a cylindrical container.

Let's see how.

So we've got our cylindrical container.

If we put a cone into that container where the circular base of the cone is equal to the circular cross section of the cylinder, and both the cone and cylinder have the same height, we can then look at what happens when we pour water into the cylinder.

And if we pour that water until the water's at the very top of the cylinder like this, we can then think what would happen if we remove the cone now from the water.

Pause the video while you think about what do you think will happen to the height of the water in the cylinder when the cone is removed, and press play when you're ready to continue.

Well, if we remove the cone from the cylinder, then the height of the water will decrease.

A bit like when you get out the bath, the height of the water decreases in the bath.

So let's do that with a cone and a cylinder.

We take it out and we can now see that the height of the water has decreased.

The height of the water is now 14 centimetres, which means it's decreased by seven centimetres.

And seven centimetres is one third of 21.

So that means the amount of water in the cylinder has decreased by one third of it's original mount.

Now that empty space at the top of the cylinder, that is the same amount of space that the cone used to take up inside the cylinder before we took it out.

Therefore, the empty space at the top of the cylinder has the same volume as the cone that was removed.

That means the volume of the cone is one third the volume of a cylinder that has the same height, and the same radius.

So let's start thinking now about formulas.

The formula for the volume of a cylinder is v, for volume, equals pi times radius squared times height.

Pi r squared h.

So if the volume of a cone is one third the volume of a cylinder, that means the formula for the volume of a cone would be v equals one third times pi times radius squared times height, where the pi r squared h is the volume of a cylinder, and we have a third of it at the front.

So let's check what we've learnt.

Here, we have a cone inside a cylinder.

The cone and cylinder show the same radius, and the same perpendicular height.

The volume of the cylinder is 960 centimetres cubed.

So what is the volume of the cone? Pause video while you write down an answer, and press play when you're ready to see what the answer is.

Well, the volume of the cone would be one third the volume of this cylinder, one third of 960 centimetres cubed is 320 centimetres cubed.

Here we've got a paper cup, which is cone-shaped, and we have a coffee mug.

Now, the paper cup and the coffee mug have the same radius, and they have the same height.

If we pour water into the paper cup, which is a cone shape, and then pour that into the coffee mug, how many full paper cups of water can be poured into this cylindrical coffee mug? Pause video while you write it down and press play when you're ready for an answer.

Well, the answer is three.

And let's think of some explanations why.

The height and the radius of both the paper cup, and the coffee mug are the same.

If they weren't the same, then it wouldn't be three, but they are the same.

The cylindrical coffee mug has three times the volume of the cone-shaped paper cup.

That means three full paper cups of water can be poured into the coffee mug.

In other words, after three paper cups, the coffee mug will be full.

So now we know that the volume of a cone is one third, the volume of a cylinder.

Let's use that fact to find some volumes of cones.

The volume of a cone can be found by first calculating the volume of the cylinder it's in, and then thinking about how the volume of a cone is one third of that.

So here we have an example of a cone inside a cylinder.

The radius of each is 11 centimetres, the height of each is 27 centimetres, and we also have a slanted length for the cone of 29 centimetres.

Which of those measures do you think we'll use in our calculations of volume? Let's take a look.

Let's first find the volume of the cylinder by using the formula v equals pi r squared h.

If we substitute in our numbers, we'd have v equals pi times 11 squared, whereas 11 is the radius, times 27.

If we then work that out in terms of pi, we'd have 3,267 pi centimetres cubed.

So when we find the volume of a cone, we'll find a third of that, which gives you 1,089 pi centimetres cubed.

Now, you may have noticed that we haven't used the 29 centimetres in any of these calculations.

The slanted length or 29 centimetres is not used when finding either the volume of the cylinder or the volume of the cone.

And this is because the slanted length is not perpendicular to either the radius, or the vertical height of the cylinder or cone.

When we perform calculations that find the areas of shapes, the calculation involves the multiplication of two perpendicular lengths, like base and height.

When we find the volume of a solid, the calculation involves the multiplication of three perpendicular lengths.

For example, when finding the volume of the cylinder in this case, we'll do pi r squared h.

So we'll do the radius times another radius, which is perpendicular to it, times the height, which is perpendicular to both of those.

But there's 29 centimetres, which is the slanted length, that is not perpendicular to any of those measures.

So that's why we don't use it in the volume formulas.

So let's check what we've learned.

Here, we have a cylinder and a cone that show the same radius and the same vertical height.

And you can see some lengths are displayed on the diagram.

In a moment, we'll find some volumes.

But to begin with, let's look at this calculation you can see here for finding the volume of the cylinder, fill in the blanks in that calculation.

Pause video while you do that, and press play when you're ready to see the answer.

Well, we have v equals pi times 48 squared, where 48 is the radius times 55, which is the height.

So complete each sentence, and give your answer in terms of pi, the volume of the cylinder is, and the volume of the cone is.

Pause video while you do that and press play when you're ready for answers.

The volume of the cone is 126,720 pi centimetres cubed.

And that's what you get when you perform the calculation we wrote down earlier.

And that means the volume of the cone would be one third of that, which is 42,240 pi centimetres cubed.

Here's another cylinder with a cone inside.

The cylinder and cone share the same circular base, and vertical height.

So complete each sentence and give your answers as decimals, so three significant figures, pause video while you do that, and press play when you're ready for answers.

The volume of the cylinder is 44,600 centimetres cubed after you've rounded it to three significant figures.

And then the volume of the cone would be one third of that, which would be 14,900 centimetres cubed.

Now, I'm hoping that when you found one third the volume of the cylinder, you found one third of it's volume before you rounded it so that your answer for the cone is accurate as possible before you perform any more rounding.

In this particular case, you get the same answer either way, but that's not always the case, so be careful.

So do we always need to have a cylinder when we want to find the volume of a cone? Well, no, we don't.

The volume of a cone can be calculated without needing the use of a cylinder for reference, and that's by using the formula we can see on the screen here, v equals one third times pi r squared h, where the pi r squared h is the volume of the cylinder, and the third gives you the volume of the cone.

We can just substitute numbers straight into that formula there.

So for example, with the cone we can see on the screen here, we have a radius of 71 centimetres, we have a perpendicular height of 150 centimetres, and a slanted length of 166 centimetres.

We're not going to use all three of those measures in our calculation for volume of a cone.

Can you see which numbers we will use? Well, let's substitute them in.

We're going to do v equals one third times pi times 71 squared, where 71 is the radius, times 150, which is the perpendicular height.

We're not using the 166, because that's not perpendicular to the others.

That gives us v equals one third times pi times 756,150.

And then if we simplify it, we get 252,050 pi, and we could leave our answer in terms of pi, and that'll be the most accurate way to write our answer, or we could write it in this form, 792,000 centimetres cubed when round to three significant figures.

So let's check what we've learned.

Here, we have a cone, but there's no cylinder this time.

You've got some measurements on there.

Could you please fill in the blanks in the formula to find the volume of the cone? Just start by filling the blanks, and we'll find the volume a bit later.

Pause video while you do that and press play when you're ready for answers.

v equals one third times pi times nine squared, because nine is the radius, and that's half of the 18, which is the diameter, times 40, which is the perpendicular height.

We don't use the 41 centimetres, because that's the slanted height.

Therefore, could you now please find the volume of the cone, and give your answer rounded to three significant figures? Pause video while you do that and press play when you're ready for an answer.

The answer is 3,390 centimetres cubed.

Okay, it's over to you now for task A.

This task contains five questions, and here are questions one and two, pause video while you work through these, and press play when you're ready for more questions.

And here are questions three and four, pause video while you work through these, and press play when you're ready for question five.

And here is question five, pause video while you do this, and press play when you're ready to see some answers.

Okay, let's go through some answers.

In question one, we've got a cylindrical mug, which is filled with water, and we've got cone-shaped paper cups, which have the same radius and same height.

And we're pouring water from the mug into the paper cups.

Now we could pour a little bit of water into each paper cup, but what this question's asking for is, "What is the maximum number of empty paper cups that we could have?" We could find that out by thinking about if we filled each paper cup with water.

And if we did that, we could fill three of those cone-shaped paper cups, because each cone is one third the volume of the cylinder, and that would leave two empty cups.

So the answer is two.

And in question two, you've got your cylinder where you're given the volume of, and you have to find the volume of the cone that has the same radius and height.

You do that by dividing the volume of the cylinder by three, and that would give 410 cubic inches.

And then in question three, you're given the volume of a cone this time, you have to find the volume of cylinder.

You find that by multiplying by three to get 10,836 millimetres cubed.

And question four, you need to find both the volume of the cylinder and the cone.

You'd get 3,840 pi centimetres cubed for the cylinder, and you would get 4,020 centimetres cubed for your cone.

And then question five, you had some cones to find the volume of, and then you had to write them in ascending order starting with the smallest.

So let's start by finding the volumes of the cones.

A, the volume would be 2,700 pi centimetres cubed.

For B, it'll be 4,116 pi centimetres cubed.

For C, it'll be 1,344 pi centimetres cubed.

And for D, it'll be 2,420 pi centimetres cubed.

So once you've got those, when you write them in ascending order, you'd put C, D, A, and B.

Well done so far.

Now, let's move on to the second part of this lesson, which is using the volume of a cone.

To find the volume of a cone, we use the radius, and we use the perpendicular height, but we don't use the slanted length.

So what do we do in situations where we know the slanted length, but we don't know the perpendicular height? How do we find the volume of a cone? Well, the perpendicular height of a right cone can be found if we are given the slanted length and the radius of the cone.

And we do that by using Pythagoras' theorem before we find the volume of the cone.

That's because l, h and r form a right-angle triangle, and Pythagoras' theorem can be used to find any one of those when one is missing and the other two are known.

And that's by using the formula like this, l would be the hypotenuse and r and h would be the shorter sides.

So let's look at an example.

Here we have a cone where we have a radius of 12 centimetres.

We have the slanted length of 29 centimetres, but we don't know the height.

Let's find the height by using Pythagoras' theorem.

The perpendicular height can be found by doing l squared equals h squared plus r squared.

And then we can substitute in the numbers that we know.

We'd have 29 squared equals h squared plus 12 squared.

We would need to rearrange this, and then simplify by calculating what we know, and then we can square root both sides to get h equals 26.

4 when round to three significant figures.

And now, we know the perpendicular height of the cone, and we already knew the radius, we can calculate the volume by doing a third times pi times 12 squared times root 697.

Now, you might wonder why are we multiplying by root 967 rather than multiplying by 26.

4? Well, the 26.

4 was a number that was rounded, which means if we use that number with another calculation, it'll not quite be as accurate as we want it to be, which is why we can use root 697 in our calculation to make our answer as accurate as possible.

Or if the last thing you typed in your calculator was the square root of 697, you could always use the answer button on your calculator instead.

Either way, you get an answer of 3,980 centimetres cubed after you've rounded to three significant figures.

So let's check what we've learned.

Here, we've got a cone with some measurements given, you've got the slanted length and you've got the radius, but you don't have the perpendicular height, could you please calculate the length of the height, and give your answer accurate to three significant figures? Pause video while you do that and press play when you're ready for an answer.

You'd solve this by substituting into Pythagoras' theorem, and rearranging it and then you get an answer of 82.

0 when round to three significant figures.

So based on that, could you please calculate the volume of the cone and give your answer accurate to two significant figures? Pause while you do that and press play when you're ready for an answer.

You would calculate the volume by doing one third times pi times 32 squared times 81.

97, and hopefully you'd either have used the full number on your calculator display, the answer button on your calculator or the surd form of the answer, and then you'll get an answer of 88,000 centimetres cubed.

The perpendicular height, or the radius of a right cone can also be found if we know the volume, we need to know the volume of the cone and also one of those other lengths as well.

For example, here we have a cone where we know the radius, but we don't know the perpendicular height.

We also know the volume of the cone.

We can find the perpendicular height by using the formula, and substituting in the numbers that we know.

So we have v, which in this case will be 14,200, and we know r will be 20.

So we substitute those numbers in, we get this equation here.

We can rearrange this equation by multiplying both sides by three to cancel out the third.

And then we have pi times 400 times h, we just want the h, so we can divide both sides by pi times 400, or we can divide by 400 pi or we can divide by each of those parts separately if you want to.

And once we get that, we'd have h equals 42,600 divided by the product of pi and 400, which will give you 33.

9 centimetres when round to three significant figures.

Let's check what we've learned.

Here, you've got a cone where you're told the radius, and you're told the volume, and you're given the formula for how to find the volume of a cone.

Could you please fill in the blanks by substituting the values from the cone into the formula? Pause while you do that and press play when you're ready for answers.

Okay, we have 1,180, which is the volume of the cone, equals a third pi times seven squared, where seven is the radius, times w.

W is the unknown perpendicular height.

So could you use that please to solve this equation, and find the length of the perpendicular height given your answer accurate to two significant figures? Pause while you do that and press play when you're ready for an answer.

The answer is 23.

Here's another question.

This time you have the volume of the cone and the perpendicular height, and you need to find the radius.

Could you please substitute all the known information about this cone into the formula that you can see on the screen, and produce an equation that involves r squared.

We'll worry about solving r in a minute, just start by writing down the equation, pause video while you do that and press play when you're ready to see the equation.

Here's our equation.

5,401, which is a volume of the cone, equals a third times pi times r squared, because we don't know what r is, times 39.

So now we have our equation.

Could you please solve it to find a length of the radius, and give your answer accurate to three significant figures? Pause while you do it and press play when you're ready for an answer.

Okay, here is the equation rearranged to have r squared as a subject.

When you calculate that and square root, you'll get 11.

5 when round to three significant figures.

Okay, it's over to you now for task B, this task contains six questions, and here are questions one and two, pause video while you do these, and press play when you're ready for more questions.

And here are questions three and four.

Pause while you do these and press play for more questions.

And here are questions five and six, pause while you do these and press play when you're ready to look at answers.

Okay, let's take a look at some answers.

For question one, you need to find missing lengths.

P would be 63.

4, and T would be 85.

0.

And both of those could be worked out using Pythagoras' theorem.

However, with t, Pythagoras' theorem would give you the radius, you'd need to double it to get the diameter.

For question two, you need to find the volumes, but you're not given the measurements you'd need.

You need to use Pythagoras' theorem first, and then you can calculate the volume, and you'll get these values here.

And question three, the volume of the cone is 1,089 pi.

By constructing and solving the equation, you need to find the length w and w is the diameter, so we could first find the radius by using this formula here, and substitute our values in to get an equation, and that would give 11 for our radius.

So we need to double it to get the length for the diameter, which would be 22 centimetres, so w must be 22.

Then question four, you're given the volume of the cone, and need to find length u and v, where u is a perpendicular height, and v is the slanted length.

Well, you would start by using the volume of the cone to work out the value of u, the perpendicular height.

And if you did that, you would get 32.

7 centimetres.

And then you can use the perpendicular height with the radius and Pythagoras' theorem to work out the value of v, and that will give you 42.

4 centimetres.

With question five, you've got a paper cup, which is cone-shaped, and you've got a cylindrical mug as well.

And you can see that the height of the mug is double the height of the paper cup, but they both have the same radius.

You need to work out how many full paper cups of water can be poured into the tall coffee mug.

Well, one way we can do it is by finding the volume of the paper cup and the volume of the coffee mug, and then doing a division to see that six paper cups full of water will fill the coffee mug.

An alternative way to solve this could be by using geometrical reasoning.

A mug with a height of nine centimetres would fit three paper cups of water.

Since then, the paper cup, and the shorter mug would have the same height of nine centimetres and the same radius.

That would mean that a mug which is double the height, would fit double the number of paper cups in, so it fits six paper cups worth of water in.

And then we have a similar question again, but this time we have a mystery coffee mug where we don't know the radius or the height, but they are expressed in algebra.

We have the radius as 5y centimetres, and we have the height as 12x centimetres.

And what you need to do is do a similar thing again.

Well, we could work out the volume of the paper cup, and that'd be 100 pi centimetres cubed, but we can't work out the volume of the mystery coffee mug.

We can however, write an algebraic expression, and then when we do a division that time we can get our answer of 3xy squared paper cups full of water will fill the coffee mug.

Fantastic work today.

Now, let's summarise what we've learned.

The volume of a cone can be calculated through displacement using a cylinder with the same perpendicular height and radius as the cone.

If a cone and a cylinder have the same perpendicular height, and radius as each other, then the cone has one third the volume of the cylinder.

The formula for the volume of a cone is v equals one third times pi r squared h, where pi r squared h is the volume of the cylinder.

So we are finding a third of that, and the lengths on a cone can be calculated using Pythagoras' theorem, or by solving an equation when you are given the volume.

Well done today, have a great day.