Lesson video

Hello, I'm Mr. Gratton.

Welcome everyone, and thank you for joining me in this lesson on the volumes of 3D shapes.

Today we will be looking at how to find the volume of a cylinder, as well as different problems involving volumes of cylinders.

Pause here to have a quick look at some of the keywords that we'll be using in this lesson.

First up, let's look at methods to calculating the volume of a cylinder.

Both cylinders and prisms are pretty similar to each other.

They do share some similar properties, but aren't exactly the same.

And also have some properties that are different from each other.

They are both three dimensional shapes, and have a cross-section that is uniform through the entirety of one length of the entire shape.

In the case of a prism, this cross-sectional shape is a polygon.

Whilst in the cylinder, this cross-section is a circle, which is not a polygon.

As Sam says, because both shapes have this constant cross-section, whether a polygon or not, we can find the volume of a cylinder in a very similar way to a prism.

This cross-sectional face that is congruent throughout the entirety of the shape is like a slice.

And stacking these slices together, like so, is like giving the prism or the cylinder a height.

And we can use this height to find the volume of a cylinder.

Well, to begin with, a volume is the space inside of a 3D shape.

The volume of a prism is the area of this constant cross-sectional face, multiplied by the height or the depth of that prism.

If the area of a cross-section is 110 centimetres squared and the height is 13 centimetres, then the volume is 110 multiplied by 13, giving 1,430 centimetres cubed for the volume.

It is always super important to calculate or identify the area of this cross-sectional face first before considering and using the height.

Focus on getting as much information as possible about the 2D face first before considering the entirety of the three dimensional shape, whether that is a cylinder or a prism.

And because the cross-sectional face of a cylinder is a circle, we can focus on this two dimensional circle before considering the entirety of the three dimensional cylinder.

The area of this circular face is pi times R squared.

And so, the volume of the 3D cylinder is pi R squared multiplied by that height, which we can call H.

Okay, here's an example.

We have a cylinder, but first, let's focus on that 2D circular face.

Its radius is eight centimetres, and the height of the 3D cylinder is 13 centimetres.

Using the formula for the volume of a cylinder, we have pi times eight squared for the area of the circular face, multiplied by 13 for the height of the cylinder.

Remember to use the order of operations.

We must first of all square the radius, where eight squared is 64.

And then we can multiply these two values together to get a volume of 832 pi, if we are asked to leave it in terms of pi.

Or we can use a calculator to convert this volume into a decimal, which we need to round.

For example, rounded to three significant figures at 2,610 centimetres cubed.

Okay, here's a check for you.

Pause here to state true or false for the statement, a cylinder is a prism.

A cylinder is not a prism.

Pause here to choose your justification for why.

The justification is all about polygons.

Both have uniform cross-sections, but a prism's cross-section is a polygon, which a circle is definitely not.

For this check, pause here to identify the correct calculations and formula for the volume of this cylinder.

A is the formula for the volume of any cylinder.

Substituting a radius of 10 centimetres and a height of 12 centimetres gives D.

And E shows the first step of calculation, the squaring of the radius.

The volume in terms of pi is 1,200 pi centimetres cubed.

Next up, here we have two steps in the process to calculate the volume of this cylinder.

Pause here to figure out what values go in the boxes A to D.

We square the radius, so A equals five.

And therefore, B is the height of 20.

We evaluate the square, five squared equals 25, and D remains 20.

Now that we have some calculations, use them to find the volume of this cylinder.

Pause now to do this.

The volume is 1,570 centimetres cubed.

And lastly, we've been given different information this time.

Pause here to find the volume of this cylinder.

The volume of any cylinder or prism is the area of the cross-section multiplied by its height.

We are given the circular cross-section at 55 inches squared.

So we multiply this by the height of six, giving us 330 inches cubed.

Sometimes on that circular base, we are given the length of the diameter, not the radius.

If that happens, halve the diameter to get the radius, and we do this before doing anything else.

Here, the diameter is 20 centimetres.

And so, halve it to get a radius of 10 centimetres.

We can use this radius of 10 centimetres to either find the area of the circular face, or the volume of the entire cylinder.

Pi multiplied by the radius of 10 then squared, then multiplied by the height of 17, gives us 1,700 pi, or 5,340 centimetres cubed.

Right, here's a familiar check.

Pause here to figure out what goes in the boxes A to E.

The radius is four, and so the radius squared is 16.

The volume of the whole cylinder is therefore 400 pi centimetres cubed.

This shape is called a semicylinder, also known as a hemi-cylinder or a half-cylinder.

It's just a cylinder that has been cut in half.

But it has to have been cut through a diameter of its circular base, so that its cross-section is a semicircle, one that is consistent through the whole length of the shape.

A semicylinder is not a prism because a semicircle is not a polygon.

Because its cross-section is half a circle, its area is also half the area of a full circle.

The area of a semicircle is pi R squared for the area of a full circle, then halved.

Therefore, the volume of a semicylinder is the volume of a cylinder, but halved.

The method for finding the volume of a semicylinder is very similar to that of a full cylinder.

In this example, we have a radius of three.

And so, the radius squared is nine.

Nine multiplied by 10 multiplied by pi is the calculation you'd need to do for a full cylinder.

So half of nine times 10 times pi, giving 45 pi, is the calculation you need for the volume of a semicylinder.

Which is 141 centimetres cubed when rounded to three significant figures.

Right, pause here to identify the correct formula and calculations for the volume of this semicylinder.

A, B, and D are the answers, giving a volume of 270 pi centimetres cubed.

And lastly, with no help given, pause here to calculate the volume of this semicylinder.

To three significant figures, we have 1,810 centimetres cubed.

Amazing stuff.

Here is question one of the practise task.

Pause here to fill in all of the blanks to find the volume of each cylinder.

And question two, here are three different cylinders.

Pause here to find out which ones have a volume of less than 7,000 centimetres cubed.

And lastly, question three, pause here to, starting with the smallest, put these semicylinders in order of the size of their volumes.

Good effort on all of those questions.

Here are the answers for question one.

Pause here to compare all of your calculations to the ones on screen.

The volumes for A and B were 11,400 centimetres cubed, and 15,100 feet cubed respectively.

And for question two, both B and C had their volumes at less than 7,000 centimetres cubed, at 6,840 centimetres cubed, and 6,960 centimetres cubed respectively.

However, A was too large, with a volume of 7,430 centimetres cubed instead.

And for question three, B had the smallest volume at 924 centimetres cubed.

Then A at 982 centimetres cubed, whilst C was the largest at 1,080 centimetres cubed.

Now that we've seen how to calculate the volume of the cylinder, let's use these volumes in a range of different problems. Let's have a look.

The formula volume equals pi R squared multiplied by the height has three variables.

The volume, the radius, and the height, or the depth of the cylinder, where pie is a constant of approximately 3.

14, not a variable length.

If you know any two of these three variables, then you can substitute them into that formula to find the value of the third currently missing variable.

Okay, let's have a look at two different cylinders.

I will model a method on the left, and then you can use my model methodology to give the question on the right a go.

The question is find the height of the cylinder, given its radius and volume.

The three variables are the volume, the radius, and the height.

I know the volume of 2,262, and the radius of six.

I carefully substitute these two values into the formula, keeping the height as H, as it is currently unknown.

I evaluate six squared, giving 36 pi multiplied by H on the right hand side of the equation.

I then divide both sides by 36 pi, which solves for H.

The height of this cylinder is the fraction 2,262 over 36 pi, which is 20 when rounded to two significant figures.

The height of this cylinder is 20 millimetres.

Pause here to try this yourself on this right hand cylinder.

18,100 is the volume, and it equals pi multiplied by 15 squared multiplied by the height, where 15 squared is 225.

We divide through by 225 pi, leaving the height at approximately 25.

6 millimetres after being rounded to three significant figures.

Right, pause here to identify the correct calculations to find the radius of this cylinder given its height and volume.

C is the formula after the volume and height are substituted in.

Whilst E is the calculation for R squared, after we have divided the volume of 925 by six pi, which is the height multiplied by pi.

Using the information from that last question, pause here again to calculate the length of the diameter of this cylinder.

We square root this expression, giving a radius of seven centimetres.

Therefore, the diameter is double that at the length 14 centimetres.

This cylinder fits perfectly inside this cuboid.

This means that this cuboid is the smallest possible cuboid that can fit this particular fixed-size cylinder.

The two faces on this cuboid that touch the circular faces of the cylinder must be squares.

This means that each circular face of the cylinder has four edges of the cuboid that are tangent to it.

The diameter of the cylinder is equal to the edge length of the square face on the cuboid.

Be careful, this is true for the diameter, not the radius.

Okay, but why should I care? What can we do with these two shapes? Well, if the volume of the cylinder is 49,000 centimetres cubed, and the diameter of the cylinder is 30 centimetres, we can find the volume of the cuboid.

Okay, let's break this down into important chunks.

To find the volume of the cuboid, we first need to find the height of the cuboid, which is also the height of the cylinder.

Since we're finding the height of the cylinder, when given the volume of the cylinder and the radius of the cylinder at 15 centimetres, we know the value of two of the three variables of the cylinder.

We can then substitute them into the volume formula to find the missing variable, that shared height of the cylinder and cuboid.

The height is approximately 69.

32 centimetres.

However, I recommend leaving this height as either a fraction or a decimal, with several decimal places to preserve as much accuracy and precision as possible.

We then find the volume of the cuboid using the two edge lengths, where the width and the depth of the cuboid are both 30 centimetres, since the base face of the cuboid is a square.

The volume of the cuboid is therefore 30 multiplied by 30, multiplied by that shared height that we just calculated, giving a volume of the cuboid of 62,400 centimetres cubed when rounded to three significant figures.

Right, for this check, pause here to identify the correct statements for this cylinder and cube.

Because we have a cube, not some general cuboid, the diameter of the cylinder is equal to all edge lengths of the cube, including its height.

And because the radius is half the diameter, the radius is half of any one of the edge lengths of the cube.

Okay, final check.

The volume of this cube is 64,000 centimetres cubed.

Pause here to find the volume of the cylinder.

If the volume of the cube is 64,000, then one edge length is the cube root of 64,000, which is 40.

One edge length is 40 centimetres.

Therefore, the height of the cylinder is also 40 centimetres.

The diameter of the cylinder is, once again, 40 centimetres.

And therefore, the radius of the cylinder is half of that, at 20 centimetres.

Now we know the height and the radius of the cylinder, we can find its volume at 16,000 pi centimetres cubed.

Brilliant, let's put our problem solving skills into practise.

Pause here for question one, where you have to find the height or radius of each of these two cylinders by first substituting all of the known information into the volume formula.

And then solving an equation.

For question two from the given information, find the height and radius of this cylinder.

For question three, a cylindrical tin fits inside this cuboidal box, so that five faces of the box are touching the cylinder.

Calculate the maximum number of tins that can fit inside this box.

Pause here to do these two questions.

Great stuff, onto the answers.

For question one, pause here to compare your calculations to these on screen, where A has a height of 30 metres, and B has a radius of 16 yards.

For question two, the height of the cylinder is the volume divided by the area of the circular face at 36.

5 centimetres.

The radius of the cylinder is found using pi R squared equals the area of the circular face, giving a radius of 29.

5 centimetres.

And finally, question three, let's focus on the cylinder to begin with.

We know its height and volume, so we can substitute this information into the volume formula to find the radius of the cylinder.

The radius is 70 millimetres.

And so, its diameter is 140 millimetres.

This diameter of 140 millimetres is shared by the edges on the base face of the cuboid.

Therefore, the volume of the cuboid is 140 for the width, multiplied by 140 for the depth, multiplied by the unknown height, giving a volume of 2.

8 million cubic millimetres.

Dividing 2.

8 million by 140 squared gives a height of approximately 142.

9 millimetres.

The height of one tin is 38 millimetres.

Dividing the height of the box by the height of the tin gives 3.

759 tins that can fit inside the box.

However, you cannot get 3.

759 tins.

This means that only three full tins can fit inside the box, as we have to round down.

Blimey, amazing work everyone on this lesson, where we have compared the properties of a prism to that of a cylinder, including comparing methods to find their volumes.

The volume of a cylinder is the area of a circular face multiplied by its height, where pi R squared is the area of that circular face.

There are three variables in the formula for the volume of a cylinder, the volume itself, the radius, and the height.

We can rearrange an equation with these known values of two of the three variables, in order to find the value of the third currently unknown variable.

Once again, everyone, amazing work on this lesson about cylinders.

I have been Mr. Gratton, and you have all been amazing.

Have a great rest of your day, and good bye.