Lesson video

Hello there and thank you for choosing today's lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

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

This lesson is called Volume of Composite Solids and by the end of today's lesson we'll be able to calculate the volume of a composite solid.

Here are some previous keywords that you may be familiar with and we'll be using again in today's lesson.

We'll revisit these words very shortly, but feel free to pause video if you want to read these definitions before pressing play to continue.

The lesson is broken into two parts, and the first part we're going to look at composite solids constructed with cuboids.

Let's begin by recapping some key definitions.

A solid is a shape that has three dimensions.

For example, width, height, and depth could be those dimensions.

And we have a picture of a solid on the screen here.

It's a cuboid.

A compound shape is a shape that is created using two or more basic shapes and a composite shape is an alternative for a compound shape.

So the example we can see on the screen here is made by joining two cuboids together that makes a compound solid or a composite solid.

The volume is the amount of space occupied by a solid, so this composite solid is constructed with two congruent cuboids.

All lengths are given in centimetres and we're going to calculate its volume.

Andeep is gonna help us.

Andeep says it could be helpful to write on some of the missing lengths before we start.

Feel free to pause video if you want to work those out for yourself before we begin and press play when you're ready to continue.

Here they are, let's now continue.

Andeep says, I may not necessarily need all of these, but they may help.

It doesn't hurt to work out more information than you need, so it's always worth doing so if they can.

Andeep says it's a prism, so I'll find the area of the cross section and multiply by the depth.

Well the cross section is a compound shape, which looks like an L shape and it's made from two rectangles.

We can find the area by splitting it into two rectangles and doing three times one for the area one rectangle and three times one for the area of the other rectangle, and adding them together.

I get six centimetres squared and then we can find the volume by multiplying by the depth.

The depth is two centimetres.

So do six, which is the area of the cross section multiplied by two, which is the depth, to get 12 centimetres cubed.

Let's check what we've learned there.

Here we have a composite solid, which is a prism constructed with two congruent cuboids and all lengths are given in centimetres.

And let's start by finding the area of the cross section of this prism.

Pause video while I do that and press play when you're ready for an answer.

Your answer is 24 centimetres squared.

And you can either do it by finding the area of each rectangle, six times two and six times two and adding them together or noticing that you have two lots of the same rectangle, so you're doing six times two, then multiplying it by two.

So now we know the area of the cross section is 24 centimetres squared.

Find the volume of the solid.

Pause video while you do it and press play when you're ready for an answer.

The answer is 72 centimetres cubed.

We use 24, which is the area of the cross section, and we multiply it by three, which is the depth.

Here's the composite side that we found the volume for earlier and earlier we found the volume of this by finding the area of the cross section and multiplying by a depth.

But what can be more fun than find the volume using one method is to find the volume using multiple methods.

And Sofia says an alternative method could be to break it into two cuboids like this.

She says, I could find the volume of each and then add them together.

Well, the volume of one cuboid would be three times two times one, and I'll get six centimetres cubed.

So to find the volume of two cuboids, we could then add them both together, six plus six or six times two, and that'll get 12 centimetres cubed.

So let's check what we've learned.

Here we have a cuboid.

Find the volume of this cuboid.

Pause video while I do this and press play when you're ready for an answer.

The answer is 36 centimetres cubed.

So let's now make this into a composite solid.

Here I have a composite solid constructed with two congruent cuboids.

We know that the volume for each cuboid is 36 centimetres cubed.

Find the total volume of the solid.

Pause video while do it and press play when you're ready for an answer.

The answer is 72 centimetres cubed.

It's the same solid we had previously, but we worked out the volume in a different way by finding the volume of each cuboid and multiplying it by two or adding them together.

So here's our composite solid again.

How else could we find the volume of it? Sam says, another method could be to complete the smallest cuboid that would surround the solid.

In other words, fill in that gap in the top left corner of this composite solid to make it into one single cuboid.

Sam says, I could find the volume of the surrounding cuboid, find the volume of the empty space and then subtract.

So the volume of the surrounding cuboid will be four times three times two, and that will give 24 centimetres cubed, but the volume of the composite solid is less than that.

So the volume of the empty space, which is also a cuboid, that would be three times two times two that would give 12 centimetres cubed and then we can subtract to get 12 centimetres cubed for the volume of the solid.

So let's check what we learned with that method.

Here we have a composite solid, which is a prism constructed with two congruent cuboids.

All lengths are given centimetres and what we're going to do first please is find the volume of this small smallest cuboid that could surround it and the dash lines that help to see what that cuboid would look like.

Pause video while you do it and press play when you're ready for an answer.

The answer is 144 centimetres cubed.

Now this time, let's find the volume of the cuboid that would fill that empty space that you can see on the screen there.

Pause video while you do it and press play when you're ready for an answer.

The answer is 72 centimetres cubed.

So if we know the volume of the surrounding cuboid is 144 centimetres cubed and the volume of the empty space is 72 centimetres cubed, could you please find the volume of the solid itself? Pause video while you do that and press play when you're ready for an answer.

The answer is 72 centimetres cubed, which you get from subtracting 72 from 144.

Here's our composite solid again.

There couldn't possibly be another way to find a volume of this, could there? Jacob says, another method could be to rearrange the two parts to make a single cuboid and then find the volume of that.

What does it mean by that? Well, don't forget that this composite solid is made from two congruent cuboids.

If we take those cuboids apart and put them back together again, it could look something a bit like this.

Now on the diagram we can see the measurements for one cuboid.

The total length of both those cuboids will be six centimetres.

That means we can find the volume of that cuboid and it'll be the same as the composite solid.

So that would be one times two times six, which would be 12 centimetres cubed.

So let's check what we've learned with that method.

The composite solid we can see on a screen is a prism constructed with two congruent cuboids.

It is rearranged to form a single cuboid like you can see in the diagram, but which calculation would find its volume? Pause video while you choose answer and then press play when you're ready to see what it is.

The answer is A, the height of the overall cuboid Once those two smaller cuboids are put together is 12 centimetres and that's from doing six multiplied by two.

Then we can do 12 times two times three to get the volume of the new cuboid that we've made, which is equal to the volume of the composite solid.

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

This task contains two questions and here is question one.

You've got two composite solids constructed with congruent cuboids and all lengths are given in centimetres.

Find the volume of each.

Pause video while you work those out and press play when you're ready for question two.

And here is question two.

It all involves a composite solid that is constructed with two cuboids, but there are slight differences between each part of this question.

Pause video while you work these out and press play when you're ready for answers.

Okay, let's go through some answers.

So question one, part A, the answer is six centimetres cubed.

And part B, the answer is 96 centimetres cubed.

And question two, this composite solid is constructed with two cuboids.

We need to find a volume in part A and that would be 110 centimetres cubed.

And we get that by fan volume of each cuboid and adding them together.

But in part B, the smaller cuboid from part A is dropped down into the large cuboid until the bases are level of each other.

Wonder how far that would've gone? Well, the height of the larger cuboid at the bottom is one centimetre, which means when we drop that smaller cuboid into it, it'll go one centimetre down.

The height of the smaller cuboid was originally three.

So if it's dropped down one centimetre, there'll be two centimetres sticking out of the top.

So we can now find the volume which we can get by doing 80, which is the volume of the larger cube we worked out earlier, add the volume of this small cube that you can now see, which is two times two times five, and altogether that will give 100 centimetres cubed.

And then part C, the two cuboids are attached at the corners as shown by removing a corner from the larger cuboid and what we need to do is find the volume of the new solid.

Well, from part A, the volume of the small cuboid was 30 centimetres cubed and from part A the volume of the large cuboid was 80 centimetres cubed.

We need to find the volume of that overlap and then subtract it.

Well, the 3D overlap forms a cuboid with the, I mean think of it as the corner that got removed from the large cuboid and that has the dimensions of 2.

1 centimetres by one centimetre by 1.

5 centimetres and that means the volume of that overlap will be 3.

15 centimetres cubed.

So the volume of the newcomers at solid is a sum of 30 and 80 and subtract 3.

15 to give 106.

85 centimetres cubed.

Great work so far.

Now let's move on to the second part of lesson which is looking at composite solids constructed with different shapes.

Here we have a hemisphere of a diameter of length 20 centimetres and we're going to find the volume and Lucas is gonna help us with that.

He says I could find the volume of a full sphere and then halve it to get the volume of the hemisphere.

On the bottom right of the screen we can see a formula for how to find the volume of a sphere.

So let's do that.

The volume of a whole sphere will be four thirds times pi times 10 cubed.

And that would give 4,000 over three pi centimetres cubed.

That means the volume of the hemisphere will be half of that.

So half times 4,000 thirds pi would be 2000 thirds pi centimetres cubed, or we can say 2000 with three pi centimetres cubed.

Now we could leave the volume in terms of pi and with the fraction and that would be the most accurate way to write the answer or we can convert it to 2090 centimetres cubed, which is rounded to three significant figures.

Now do we always want to be doing it in two steps like that? Finding the volume of a whole sphere and then finding the volume of a hemisphere by halving it? Could we come up with a new formula to help us work out the volume of a hemisphere? Well, we could do that by considering this question algebraically.

So here we have a hemisphere where the radius is labelled R this time and Lucas writes an expression for the volume of the hemisphere.

Well, the volume of a whole sphere is four thirds pi R cubed and then we multiply it by a half to find the volume of a hemisphere.

And that would give four over six pi R cubed and we can simplify it to get two thirds pi R cubed for the volume of a hemisphere.

Lucas says, the volume of a hemisphere can now be found by substituting the radius into two thirds pi R cubed.

But if you're worried about forgetting that formula, you can always find the volume of a whole sphere by using the formula for the volume of the sphere and then halve it afterwards.

So let's check what we've learned.

Here's a hemisphere, find its volume and give the answer in terms of pi.

Pause video while you do it and press play when you're ready for an answer.

The answer is 144 pi centimetres cubed.

And the work you can see on the screen here is how you'd do it if you found the volume of a whole sphere then halved it to get the volume of a hemisphere.

Alternatively, you could find the volume of a hemisphere by substituting six into two thirds pi R cubed and you get the same answer.

Let's now look at a composite solid which is constructed with a cylinder and a hemisphere and each have a diameter of 20 centimetres.

We're going to find the volume and Lucas is going to help us.

He says, I could find a volume of the hemisphere and the cylinder and then add them together.

So let's do that.

The volume of the hemisphere we worked out earlier is 2000 thirds pi centimetres cubed.

The volume of the cylinder in this case will be pi times 10 squared.

That's the area of the circle which we can see highlighted at the bottom of the cylinder.

And then we can multiply it by 21, which is the height of the cylinder in this case.

And that would give 2,100 pi centimetres cubed.

And then we can find the total volume by adding those two volumes together to get 8,300 thirds pi centimetres cubed, which can also write as 8,690 centimetres cubed, which has been rounded to three significant figures.

Let's check what we've learned then.

Here's a cone, find its volume in terms of pi and there's a formula on the screen to help you.

Pause video while you do it and press play when you're ready for an answer.

The answer is 120 pi centimetres cubed.

The way you get it is first work out the radius, which is six centimetres, it's half the diameter, and substitute it into that formula we can see there.

A third times pi times six squared times 10, which is the height, and we get 120 pi centimetres cubed.

Let's now turn this into a composite solid.

Here is a composite solid constructed with a cone and a hemisphere.

Find the volume in terms of pi and there are some formats on the screen to help you.

Pause video while you work it out and press play when you're ready for an answer.

The answer is 264 pi centimetres cubed.

Now the cone is the same as the one you had earlier, but if it wasn't, you'd need to work out the radius first, which we can get by doing 16 subtract 10 and that would give us six, and then we can work out the volume of the cone, work out the volume of the hemisphere and add them together.

Here's a different composite solid now, which is constructed with a cylinder and two hemispheres, and they each have a diameter of 20 centimetres.

We're going to find the volume along with Lucas.

He says, I could find the volume of each of the two hemispheres and the volume of the cylinder, so that's three volumes and then I would add together the three volumes.

Could that be done in a more efficient way? Pause video while you think about what Lucas could do differently and then press play when you are ready to continue.

Well, let's not forget that a hemisphere is half a sphere and if we have two hemispheres, they can be put together to make a whole sphere.

So Lucas says the volume of the two hemispheres will be equal to the volume of a whole sphere with the same radius.

So I could just add the volume of the sphere to the volume of the cylinder.

Let's do that.

The volume of the sphere is 4,000 thirds pi centimetres cubed.

The volume of the cylinder is 2,100 pi centimetres cubed, and then we can add them together and we would get 10,300 over three pi centimetres cubed, which we could write as 10,800 centimetres cubed, which is rounded to three significant figures.

So let's check what we've learned then.

This composite solid is constructed with two hemispheres and a cylinder which all have the same diameter, and you need to find its volume given your answer in terms of pi.

And there's a formula under screen to help you.

Pause video why do it and press play when you're ready for an answer.

The answer is 576 pi centimetres cubed.

To get that, you could do 20 subtract eight to get the diameter of the circle, which is part of each of the three parts, and that'll be 12.

And then half of the radius, which is six.

If you wanna just double check the six is right, you can think about how the distance from the top of the solid toward where a cylinder starts is six, then add an eight for the cylinder, then add another six to the bottom of the solid, and that would give 20, which is what it says is the total height of the overall composite solid.

So now you know that the radius is six, sou can substitute into your formulas, work out the volume of each part and an atom together to get 576 pi centimetres cubed.

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

This task contains five questions and here are questions one to three.

You have some composite solids and you have some formulas on the screen to help you find them volumes of them.

Pause the video while you do that and press play when you're ready for more questions.

And here are questions four and five.

In question four, you've got a stone basin, which is made by carving a hemisphere out of a cuboid.

So the cuboid is solid, but the hemisphere that's cut out of it is a bit of empty space.

And usually you might fill that with water or something.

The lowest point of the hemisphere is eight centimetres above the base of the cuboid and you need to find the volume of the overall basin.

And then question five, you can see you've got a cube and a cylinder which are being pushed together.

A vertex of a metal cube, which is the bit at the top, is pushed two centimetres into a polystyrene cylinder base.

The top vertex is vertically above the bottom vertex.

The cube creates an indent into the cylinder that is a right equilateral triangular pyramid with the area of the base being 10.

39 centimetres squared.

The cube and the cylinder are stuck together, so what is the volume of the composite solid formed? And if you want to, there's a link to a GeoGebra file, which allows you to look at that shape from different angles and explore it more.

There are some formulas on the screen to help you.

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

Okay, let's go through some answers.

Question one.

The volume is 48 pi centimetres cubed.

You get it by finding the volume of the hemisphere, the volume of the cone and adding them together.

Question two, the volume of the solid is 108 pi centimetres cubed, which you can get by finding the volume of the hemisphere, the volume of the cylinder, and adding them together.

And question three, the total volume of the solid is 126 pi centimetres cubed, which you can get by finding the volume of both hemispheres, which is the volume of a whole hemisphere and the volume of a cylinder and adding them together.

Question four, well the volume of the cuboid will be 13,500 centimetres cubed.

The volume of the hemisphere will be 2094.

39, and there's gonna be more decimals centimetres cubed.

So we can get the overall volume of the basin by subtracting the volume of the hemisphere from the cuboid, and it's best to use all the digits in the volume of the hemisphere when we do that.

That means we get a answer of 11,400 centimetres cubed when it's rounded to three significant figures.

Then question five.

The volume of the cube will be 1,728 centimetres cubed.

The volume of the cylinder will be 400 pi centimetres cubed, but we don't wanna use all of the volumes of both of those shapes cuz there's a bit where they overlap the indent.

That is a pyramid.

The volume of the pyramid is 6.

9266, and there are more decimals centimetres cubed.

So to find the total volume of the composite solid, we can add together the cube with the cylinder and subtract the overlap in pyramid, and that will give 2,980 centimetres cubed.

Fantastic work.

Let's now summarise what we've learned.

A composite solid can be decomposed to make the volume easier to calculate.

By decompose we can mean to break it apart into its basic shapes, find the volume of each and then add them together again.

Or completing the solid and then subtracting the volume of the empty space may be a useful method as well.

Alternatively, decomposing and rearranging the parts of a solid may provide a method to find the volume by using a simpler calculation.

Well done today.

Hope you have a great day.