Lesson video

Hello there and thank you for joining me.

My name is Dr.

Rowlandson and I'll be guiding you through today's lesson.

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 surface area of composite solids and by the end of today's lesson we'll be able to calculate the surface area of composite solids.

Here are some previous keywords that you may be familiar with already and we'll be using these words quite a lot during today's lesson.

Now we'll revisit these words very early on lesson, but feel free to pause video if you want to read the definitions on the screen and then press play when you're ready to continue.

This lesson is broken into two learning cycles and we're going to start by looking at composite solids that are constructed with cuboids.

Let's begin by looking at some definitions in detail.

A solid is a shape that has three dimensions.

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

And we can see an example of a solid on the screen here.

It's a cuboid.

A compound shape is a shape created using two or more basic shapes.

A composite shape is an alternative word for a compound shape as an example on the screen of that here, this composite shape or compound shape is made from two cuboids.

The surface area of a solid is a total area of all the surfaces and sometimes surface area is abbreviated to SA.

Here we have a composite side that is constructed with two congruent cuboids.

All the lengths we can see here are given in centimetres.

And what we will do now is we're going to calculate its surface area and Alex is going to help us.

He says it could be helpful to write on some of the missing lengths, 'cause we can see three lengths on that composite shape, but there are plenty of other lengths, where we could also write down as well.

Feel free to pause the video if you want to work out those missing lengths and then press play when you're ready to continue.

Well here they are.

Did you get them? Let's continue.

Alex says, this solid has eight faces.

Let's now label those faces A to H so that we can label our calculations according to which face we are calculating.

So this top face here, let's label A, we have another face, another face here, another face here, and another face at the front here.

Now those are all the faces that we can see, but there are also some faces on this compound shape that we cannot see.

There's a face at the back on the left and there's a face underneath and there's a face behind on the right as well.

So now we've labelled all our faces.

Alex says, I could find the area of each and then add them together and that'll find the total surface area.

Let's do that together now.

Faces A to D are rectangles, so face A would be two centimetres squared, B would be four centimetres squared, C would be six centimetres squared, and D would be two centimetres squared.

Now E and F are not rectangles, they are composite shapes, but we can split them into two rectangles, which means the area for E could be done by doing three times one plus three times one.

Each of those multiplications is the area of a rectangle.

And that gets six centimetres squared.

F is the same as E, so that's six centimetres squared as well.

G will be eight centimetres squared and H will be six centimetres squared.

So to find the total surface area, let's add up now the areas of each of those faces and we'd get 40 centimetres squared.

If you would like to pause the video and check how each of those numbers and the calculations match the lengths on the composite solid, feel free to do so and then press play when you're ready to continue.

Here we have Izzy.

Izzy says, I found the surface area in a more efficient way by splitting it into its two congruent cuboids 'cause that's how it was created.

She says, I found the surface area of one of the cuboids and then doubled it.

Let's see what she did.

The surface area of one cuboid is 22 centimetres squared, and she's done that by finding the area of three rectangles, adding them together and doubling it to get the area of all six rectangles that make up the cuboid.

So if the surface area of one cuboid is 22 centimetres squared to get the surface area of two cuboids, she's multiplied it by two to get 44 centimetres squared.

Huh? 44 centimetres squared.

Izzy says Alex got 40 centimetres squared for his answer.

Why is my answer greater than that? What is wrong with Izzy's method here and how could it be corrected? Pause the video while you think about that and press play when you're ready to continue.

Izzy says part of each cuboid is covered by part of the other cuboid and these parts will not contribute to the total surface area of the overall composite solid here.

Let's take a look at that.

If we look at the rectangle, which is now shaded in grey on the screen, that rectangle makes up part of the surface area of each cuboid individually, but it doesn't contribute to the surface area of the composite solid.

So Izzy says I could work out the area of this rectangle here and then subtract two lots of it, one for each cuboid, from 44.

So the area of the overlap is two centimetres squared.

If we subtract two lots of it from 44, yes, we get 40 centimetres squared.

Well done, Izzy.

Let's check what we've learned.

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

All lengths we can see here are given in centimetres.

Your first job with this is to find the values of A, B, C, and D.

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

Here are the answers.

A is three, B is two, C is six, and D is four.

So let's now do a little bit more with this.

Here we have just one of the cuboids highlighted.

Find the surface area of this cuboid.

Pause the video while you do this and press play when you're ready to continue.

Here's our answer.

72 centimetres squared.

You can find the area of each of the three faces you can see, add 'em together and then double it to get a total surface area.

We now have the entire composite solid and there is one rectangle where the two cuboids overlap.

Find the area of that rectangle.

Pause while you do it and press play when you're ready to continue.

The answer is six centimetres squared.

So let's now find the total surface area of the overall composite solid.

And you've got some information at the bottom of the screen, which are things that you worked out in the previous questions.

Pause the video while you work out the surface area and then press play when you're ready to continue.

The answer is 132 centimetres squared.

Now there are lots of different ways you can work that out, but one way could be to do 72, multiply by two, and then subtract two lots of six and that'll give you 132.

Let's now work through another example.

This composite solid is constructed by placing a cube on top of a cuboid and all lengths we can see here are given in centimetres.

So let's calculate the surface area.

Jun's gonna help us with this.

He says I could find the surface area of each solid separately, then I could add 'em together and subtract two lots of the overlapping area.

Let's do that together.

So the surface area of the cube would be 24 centimetres squared.

That's two times two for the area of a square and then times it by six because there are six faces to a cube.

And a surface area of a cuboid will be 58 centimetres squared, where you can find the area of three of the rectangles and then add them together and double it to get the area of all six rectangles.

And the area of the overlap, well that would be two times two, which is four centimetres squared.

So to find the total surface area, we can do 24, which is the surface area of the cube, plus 58, which is the surface area of the cuboid, and then subtract two lots of four, which is the area of the overlap, and that gives 74 centimetres squared.

Jun says if the cube was moved over here to a different part of the solid, the total surface area would still remain the same.

This is because the area of the overlap is still four centimetres squared.

He says, so long as the entire of the bottom face of the cube is connected to a face on the cuboid, the surface area will remain the same like it is here as well.

However, if the cube hangs over the edge of the cuboid, then the area of the overlap will be less than four centimetres squared.

So now the surface area overall would increase like we can see here.

Let's check what we've learned.

Find the surface area of the cube below and all lengths are given in centimetres.

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

The answer is 150 centimetres squared.

Do five times five to find the area of one square and then multiply it by six because there are six squares on the faces of a cube.

So here we have a cuboid.

Find the surface area of this.

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

The answer is 164 centimetres squared.

You can get it by multiplying pairs of lengths to find the areas of three of the rectangles, for example, the ones you can see, add them together.

And because the other three rectangle are congruent, you can then double your answer to get the overall surface area.

So here we have a composite solid, which is constructed by placing the cube you saw earlier on top of a cuboid you saw earlier.

All lengths again are given in centimetres.

You're given the surface area of the cube and the surface area of the cuboid.

What you need to do is find the surface area of the composite solid.

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

The answer is 264 centimetres squared.

Now there are lots of different ways you can work it out.

You may have worked out the area of each face separately and added 'em together.

But using information on the screen here, you can work it out by adding together the two surface areas that you have and subtracting two lots of five times five, where five times five is the area of the overlapping face.

So it's over to you now for task A, this task contains two questions, and here is question one.

You have two composite solids.

Each one is constructed by using two congruent cuboids and all the lengths are given in centimetres.

What you need to do is find the surface area of each.

Pause the video while you do it and press play when you're ready for question two.

And here is question two.

You once again have two composite solids.

Each one is constructed with cuboids and you need to find the surface area of each.

Pause the video while you do this and press play when you're ready to see some answers.

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

Well question one part A, our answer is 26 centimetres squared, and here is the working for how you can get it.

Feel free to pause the video while you check this against your own and then press play when you're ready for the next answer.

Then part B, the surface area of the composite solid is 152 centimetres squared and here's an explanation for how you get it.

Feel free to pause video if you want to check it and then press play when you're ready for another answer.

Then question two, the surface area of this composite shape is 238 centimetres squared and the working is on the screen for how to get that.

Feel free to pause video if you want to read through this and press play when you are ready for the next answer.

And part B, the surface area is 246 centimetres squared and here is the working on the screen for that as well.

Pause video while you check this against your own and then press play when you're ready to move on to the next part of today's lesson.

Well done so far.

Now let's move on to the next part of today's lesson, which is looking at composite solids that are constructed with different shapes.

Here we have a hemisphere.

The hemisphere has a diameter of length 20 centimetres, and what we're going to do is calculate its surface area.

Oh, hello, Aisha.

Aisha says, I'll start by writing a plan for my calculations.

And here's her plan.

Step one will be to find a surface area of the whole sphere.

And in the bottom right hand of this screen we have the formula for the surface area of a sphere.

Step two will be to find the area of the curved surface, which we can see here is on the top of this hemisphere.

Step three will be to find the area of the circle, which we can see is underneath the hemisphere on the screen.

And then step four will be to find the total surface area of the hemisphere.

Let's now work through that together.

So step one was to find the surface area of the whole sphere.

That would be four, multiplied by pi, multiplied by 10, which is the radius, squared, and that would give 400 pi centimetres squared.

Well if that is the surface area of an entire sphere, then the area of the curved surface would be half of that.

So we could do 400 pi divided by two, and that would give 200 pi centimetres squared.

And the area of the circle, well that would be pi times radius squared, and the radius is 10 centimetres.

So that means it'd be pi times 10 squared, which would be 100 pi centimetres squared.

And then we can find the total surface area by doing 200 pi, which is the area of the curved surface, plus 100 pi, which is the area of the circle, and that would give 300 pi centimetres squared.

Now we could leave that in terms of pi and that would be the most accurate way to write the answer, or we can convert it to 942 centimetres squared when we round it to three significant figures.

Now Aisha says this answer is equivalent to doing three pi R squared.

Is that always true or is it just the case for this particular hemisphere? What do you think? Do you think that would be always true that the surface area of a hemisphere, would be three pi R squared, or is that just the case for this particular one where the diameter was 20 centimetres? Did the numbers just work out like that? Pause the video while you think about it and then press play and we'll explore it together.

Let's now explore this together.

And one way that you can check whether something is always true or just sometimes true for particular numbers is to work through it with algebra.

So Aisha writes an expression for the surface area of the hemisphere where the radius this time is just labelled as R.

Well, the surface area of the whole sphere will be four pi R squared.

That's from the formula.

The area of the curved surface will be half of that, so that'll be four times pi R divided by two, which would be two pi R squared and the area of the circle will be pi R squared.

That's the formula for how to find the area of a circle.

So when we find the total surface area, we'd do two pi R squared, which is the area of the curved surface plus pi R squared, which is the area of the circle.

And yes, we would get three pi R squared.

So Aisha says the surface area of a hemisphere is always three pi R squared.

So that means when you find a surface area of a hemisphere, you can either break it down into parts like we did earlier and find the area of each face separately, add 'em together, or you could substitute the radius into the expression three pi R squared.

Let's check what we've learned.

Find the total surface area of this hemisphere we can see on the screen here and give your answer in terms of pi.

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

The answer is 75 pi centimetres squared.

The working you can see on the screen here is what you'd do if you found the area of each surface and added them together.

Whereas this working shows an alternative method by doing three times pi times R squared, you get the same answer either way.

Here we have a composite solid which is constructed with a cylinder and a hemisphere, and each have a diameter of 20 centimetres.

And what we're going to do now is find the surface area and Aisha's gonna help us.

She says the circle at the bottom of the hemisphere does not contribute to the surface area of the composite solid.

So we don't necessarily want to find the area of both surfaces of the hemisphere.

However, she does say that the circle at the bottom of the cylinder is congruent to it.

So that means that the curved surface at the top of this solid plus the circle at the bottom of the solid, those go together to make the total surface area of the hemisphere, even though part of the hemisphere itself is being covered by the cylinder.

So she says, I could find the area of the curved surface of the cylinder and then add it to the surface area of the hemisphere.

So let's do that together.

The surface area of the hemisphere, including the base, would be 300 pi centimetres squared.

Now to find the curved surface of the cylinder, that would be a rectangle with the height as 21 centimetres and its length would be equal to the circumference of the circle.

So we need to find the circumference of the circle, which is 20 pi centimetres.

So that means the curved area of the cylinder would be 20 pi multiplied by 21, which gives 240 pi centimetres squared.

And that means the total surface area will be the sum of these two parts here, 300 pi plus 420 pi gives you 720 pi centimetres squared.

And if you want to, we can convert that to 2,260 centimetres squared, which has been rounded to three significant figures.

So let's check what we've learned.

Here we have a cone.

You need to find the area of the curved surface of the cone, so that's not including the circle at the bottom, the curved part, and you've got the formula at the bottom of the screen for how to do that.

Give your answer in terms of pi.

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

The answer is 60 pi centimetres squared.

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

The composite solid is constructed with a cone and a hemisphere, and what you need to do is find the surface area of this composite solid, giving your answer in terms of pi, and you've got some formulas there to help you.

Pause the video while you do this and press play when you are ready for an answer.

The answer is 110 pi centimetres squared, and you can get it by finding the curved area of the hemisphere, the curved area of the cone, and then adding them together.

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

And we're going to find the surface area along with Aisha.

She says, the circle faces for each individual solid do not contribute to the surface area of the composite solid.

We can see each hemisphere has a circle and the cylinder has two circles, but none of those circles are gonna be counted in the surface area of the overall shape.

So here's her plan.

She will first, find the surface area of the whole sphere, and then she'll divide it by two to get the curved area of each hemisphere.

And then she'll find the curved area of the cylinder, and then she's going to do the curved area of the cylinder, and then she's gonna add two lots of the curved area of the hemisphere.

Hmm, I wonder if there's a better way to do that.

Aisha says, could I do this more efficiently? What do you think? Pause the video while you think about it and press play when you're ready to continue.

Well, according to Aisha's plan, she's going to find the surface area of a whole sphere and divide it by two to get each hemisphere.

And later she's gonna multiply it by two because she has two hemispheres.

Aisha says the two hemispheres make a whole sphere.

So here's her new plan.

She's going to find the curved surface of the cylinder and then add it to the surface area of the sphere.

Let's do that.

The surface area of a whole sphere is 400 pi centimetres squared, and you can get that by using the formula in the bottom right hand of this screen.

The curved area of the cylinder will be 420 pi centimetres squared.

It's what we worked out earlier.

And then we can get a total surface area by adding those together to get 820 pi centimetres squared, or we can convert it to 2,580 centimetres squared, which has been rounded to three significant figures.

Let's check what we've learned there then.

The composite solid here is constructed with two hemispheres and a cylinder, which all have the same diameter.

And what you need to do is find its surface area giving your answer in terms of pi.

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

The answer is 180 pi centimetres squared.

You first need to work out the radius of the hemisphere, which is the same as the radius as the cylinder as well, so it'll be useful for all parts.

And the way we can do that is by doing 18 subtract eight that'll give you the diameter of the sphere and divide it by two to get the radius of the sphere.

And then substitute it into all your parts of the calculations and with the working you can see on the screen.

Okay, it's over to you now for task B, this task contains five questions, and here are questions one to three, and there are two formulas at the bottom of the screen here, which can help you with these questions.

Pause video while you work through these and press play when you're ready for more questions.

And here are questions four and five.

Pause the video while you work through this and press play when you're ready for answers.

Let's go through some answers then.

Question one, the radius of the cone and hemisphere is three centimetres.

You get that by dividing the six by two.

The curved surface area of the cone is 30 pi.

The curved surface area of the hemisphere is 18 pi, which means a surface area altogether of the composite solid is 48 pi centimetres squared.

And question two, where we have a hemisphere and a cylinder, the radius of each would be three centimetres again, the surface area of the hemisphere, which includes a circle on the far right of the cylinder will be 27 pi.

The curved surface area of the cylinder will be 60 pi.

That means the surface area altogether will be 87 pi centimetres squared.

And then question three, we have two hemispheres and a cylinder where the two hemispheres make a whole sphere.

So from question two, we can get the curved surface area of the cylinder is 60 pi and the area of the full sphere as being 36 pi.

So the total area of the composite shape is 96 pi centimetres squared.

Then question four, this composite solid is constructed from a hemisphere on top of a cuboid, and the shape which will be in the overlap, which will not contribute to the surface area is a circle.

Let's find the surface area.

Well, the radius of the hemisphere is four centimetres, that's half its diameter.

And the surface area of a hemisphere can be found by doing three pi R squared, which in this case will be 48 pi.

And the area of the circular face of the hemisphere will be 16 pi.

To define the surface area of the composite shape, we're going to do 48 pi from the hemisphere plus 164 from the cuboid, and then subtract two lots of that circle, which is in the overlap, and that would give an answer of 214 centimetres squared and rounded to three significant figures.

Now in question five, once again, we have a hemisphere and a cuboid, or in this case it's a cube, but the shape on the overlap this time would be a square.

So let's find the surface area.

Well, the radius of the hemisphere will be four centimetres.

The surface area of the hemisphere will be 48 pi centimetres squared.

The surface area of the cube would be 150 centimetres squared.

And this time the area of the two joining faces would be 50 centimetres squared, which is two lots of the square.

So the surface area of the composite solid will be the 48 pi from the hemisphere plus the 150 from the cube.

Subtract 50, which is two lots of that square, which is 251 centimetres squared when rounded to three significant figures.

Fantastic work today.

Now let's summarise what we've learned.

The surface area of a composite solid is found by summing the area of all faces.

The faces may not always be the same shape though, there may also be multiple methods for finding a surface area of a composite solid, which means that surface area may well be able to be calculated in a more efficient way when we consider alternate methods.

Well done today.

Thank you very much.