Lesson video

Hello, my name is Dr.

Rowlandson and we have a great lesson in store for today.

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 checking and securing understanding of surface area of cuboids.

And by the end of today's lesson we'll be able to efficiently calculate the surface area of cubes and cuboids.

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

So you may 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.

The lesson is broken into three parts and we're going to start by looking at the surface area of a cuboid from its net.

Here we have a cuboid, which is a 3D object, and Sam says, "How can you find a surface area of a 3D object?" Well, Aisha says, "I think we could just find the area of its 2D net," like this.

Hmm, but Sam says, "How do you find the area of such a complex net?" It is quite a complex net.

It's got six faces.

Well, Aisha says, "By looking at the area of each individual rectangle first." So we can break it apart a bit like this and turn what is quite a complex problem into six smaller problems. Let's take a look at one of these rectangles.

It is 20 centimetres by 16 centimetres.

We can find the area of one rectangle by multiplying its length by its width, so in this case we'll do 20 times 16 and that would give 320 centimetres squared.

We have five more rectangles left to find the area of, could you please pause the video and find the areas of those based on these new lengths I've given you? Pause while you do that and press play when you're ready to continue.

Let's take a look at the areas.

The area of this rectangle that would be eight multiplied by 16.

We can see its length is 16 centimetres and we can see it's width is eight centimetres by comparing it to the rectangle at the bottom of the net.

Now I get 128 centimetres squared.

This rectangle, that will be eight multiplied by 20.

Now this doesn't have any length label on it, but we can get the eight centimetres because the rectangle is as wide as the one above and below it and we can get the 20 centimetres because that rectangle is as high or as tall as the rectangles on the left and right of it, that would give us 160 centimetres squared.

Now we found the area of three rectangles already and we have three left to find, but we don't need to do any calculations to find the areas of these three remaining rectangles.

That's because the other three rectangles are congruent to the first ones we found.

So the areas will be the same as the first three rectangles.

To be thorough, let's think about which areas match up with which rectangles, which areas from the numbers we've found so far, match the remaining rectangles.

Pause the video while you think about that and press play when you're ready to continue.

Here they are.

We can match them up by looking for the length and width on each rectangle.

And even we can't see it directly on the rectangle itself, we can look elsewhere across the net to see what the length or width is for each one.

And rectangles that have the same length and width are congruent and we'll have the same area.

So if we connect these six rectangles together again, they would become the net of the cuboid again and we could fold the net up to create the cuboid a bit like this.

And now we can think about what the area of each face in that cuboid would be.

This one would be 160 centimetres squared.

This one will be 128 centimetres squared.

And this one will be 320 centimetres squared, and the other three faces will be equal in area to these three we've labelled already.

If you don't necessarily label it in the right way, because sometimes it's not easy to see without the measurements on the diagram itself, doesn't matter because you'll still have all the same six areas which we're going to add up to find the surface area of the cuboid.

So if we add together all the six faces on the net, we get a calculation like this, and that would give us 1,216 centimetres squared.

So that is the surface area of the cuboid.

Let's check what we've learned.

Here, we've got six rectangles that can be connected to create the net of a cuboid.

You've got one rectangle labelled A, what is the area of that rectangle? Pause video while you write down your answer and press play when you're ready to see what it is.

Well, rectangle A doesn't have any measurements on it, but by comparing it to the other rectangles in the net we can see it would be 20 units by 11 units, so it's area will be 220 square units.

Here's the net with the rectangles joined together now and you've got two more rectangles labelled B and C.

Find the area of those.

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

Well, rectangle B, it's got a width of three units, which we can see on the rectangle itself.

The length isn't labelled on the rectangle, but we can see its length is the same as the other rectangles on its left, so it's 20 units.

That means it's area will be 60 square units.

For rectangle C, we don't have any measurements on the rectangle itself, but what we could do is look for its congruent rectangle elsewhere in the net.

That would be this one.

These two rectangles would have the same length and width that would have the same area as well.

That new rectangle we labelled, we can see it's width as three units, its length is 11 units, therefore its area would be 33 square units.

So something we can take away from this is that if a face of the net of a cuboid isn't clearly labelled.

You could see if you can calculate the area of its opposite, congruent face.

And then you know the area of the rectangle you're trying to find.

Here's another net, and you've got six faces labelled A to F.

Could you please find the area of all six faces? Pause video while you do that and press play when you're ready for answers.

Here are the answers.

Pause while you check these against your own and press play to continue.

Now we have those answers, could you please add together all six areas to find the surface area of the cuboid that this net is from? Pause video while you do that and press play when you're ready for an answer.

The surface area would be 184 centimetres squared.

Let's take a look at this question in a little bit more detail.

Sam says, "Do I have to find the area of all six rectangles? There's always going to be three pairs of rectangles.

So can't I just find the area of three unique faces and then double the sum to get the surface area?" The answer is yes, you can.

So for example, in this case, we can see we have 20 plus 12 plus 60, and because we have two lots of each of those, we could then double our answer and that will give us 184 centimetres in a different way.

You could do two times 20 plus two times 12 plus two times 60, but it'd be even more efficient to do 20 plus 12 plus 60 and then double your answer afterwards.

Either way, you will get the same answer and same as well is if you add all six faces together separately.

Let's check what we've learned with that bit.

Here you've got a net, please find the area of rectangles A, B, and C.

Pause while you do that and press play when you are ready for an answer.

Here are your answers.

A is eight centimetres squared, B is 14 centimetres squared and C is 28 centimetres squared.

So using that, could you please find the surface area of the whole cuboid that this net is from? Pause video while you do that and press play when you're ready for an answer.

The answer is 100, which we get from doing eight plus 14 plus 28 multiplied by two.

Therefore, the surface area is 100 centimetres squared.

Here's another net.

Please find the area of rectangles, A, B, and C.

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

Here are your answers.

A is 49 centimetres squared, that one's a square.

B is 140 centimetres squared, and so is C.

Now using that, could you please find the surface area of the whole cuboid that this net is from? Pause video while you do that and press play for an answer.

Here's your calculation, which means your answer will be 658 centimetres squared.

Now with this one, you may notice that you have four rectangles that are all congruent to each other.

They all have an area 140 centimetres squared.

So why have we got four rectangles this time which are congruent? That's because we have the square in the middle there because two of the opposite faces of the cuboid are squares.

The other four faces are all equal in area.

So while this cuboid has four faces which are all congruent to each other, you know it has more congruent faces? A cube.

Here we have the net of a cube.

Sam says, "There must also be a really simple calculation for the surface area of a cube since all six squares are the same size." And Sam is absolutely correct here.

Let's find the surface area.

A cube has six faces of equal area, which means to find a surface area of the cube, we could do 10 times 10, which is the area of one face, that's 100 centimetres squared, and then we can multiply it by six because we have six faces to this cube and that would give us, in this case 600.

So the surface area will be 600 centimetres squared.

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

We've got the net of a cube and you've got some calculations to choose from.

Which of these calculations are correct for the surface area of the cube that this net is from? And there may be more than one correct answer.

Pause the video while you choose some answers and press play when you're ready to see what they are.

Here are your answers.

All of these will give you an answer of 150 centimetres squared.

B, is finding the area of each face individually and adding them together.

D, is a bit like that, but you've already found the area of each face, so we're adding them together.

E, they found the area one face already, which is 25, and the times in by six and F is a single calculation, five times five, which is the area one face and then times by six in one calculation.

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

Which of these calculations are correct for the surface area of the cuboid that this net is from? Pause the video while you choose and then press play when you're ready for question two.

And here is question two.

You've got six rectangles that can be connected to create the net of a cuboid.

And you've got three questions to answer based on this.

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

And here is question three.

You've got three nets for cuboids, starting with the smallest place these nets in order of size of the surface area.

Pause while you do it and press play for question four.

And here is question four.

You've got two nets of cubes and a question to think about.

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

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

For question one, these are the calculations that'll give you the surface area of the cuboid from this net.

Then question two in part A, you had to calculate the areas of rectangles A, B, and C.

Here they are 135, 120 and 300.

Part B, you had to calculate the areas of the three remaining rectangles.

Here they are.

And then for part C, find the surface area.

You would add these together and you would get 1,110 square units.

Then question three, you had to write these nets in order of size from the smaller surface area to the greatest.

That would involve finding the surface area of each net, and you would first need to find the area of each unique rectangle in the net.

And then you've got the surface areas and then you can write them in order.

A is the smallest, then C and then B.

We've got our contextual problem about two cubes.

The net of the small cube and large cube are printed using purple ink.

And one mil of purple ink covers 600 square centimetres of area.

I want to work out how much more ink is needed to print in a larger cube than the smaller cube.

Well, you'll need to start by finding a surface area of each and here they are.

And once you've done that, we can think about how much ink is needed for the small cube by doing a division which will give 6.

25 millilitres of ink.

And then do a division for the large cube and you'd get 42.

25 millilitres of ink and then you can find the difference between them.

And that would give 36.

The large cube needs 36 mil more than this small cube.

Well done so far.

Now let's move on to the second part of this lesson, which is looking at surface area of a cuboid from plans and elevations.

Here we have another cuboid and Sam says, "Because three faces on a cuboid are just copies of the other three faces, I don't need to look at the whole net in order to find its surface area." Sam says, "I just need to find the areas of three faces: it's front face, it's side face, and it's top face." Aisha says, "Yes, you could use the three elevations to find each area, but don't forget to double the sum." So the front face will be the front elevation, the side face will be the side elevation and the top face will be the plan.

Let's take a look at this with an example.

Here we've got a cuboid and we can draw the three viewpoints accurately on grid squares.

The front elevation would be like this, three units by four units.

The side elevation would look like this.

That would be three units by five units and the plan view would look something like this.

That would be five units by four units.

And now we've done that, we can find the area of each of these faces.

The area of the front elevation will be 12 units squared.

The area of the side elevation will be 15 units squared and the area of the plan view will be 20 units squared.

Aisha reminds us that, "The other three faces on the cuboid are the same as these three." So to find a surface area we can add these three areas together and then double the sum, like this.

12 plus 15 plus 20 is the area of those three faces.

And then we'll go double it because we have three more faces just like them.

That gives 94.

So the surface area is 94 square units.

Let's check what we've learned.

Which of these rectangles show the following viewpoints of this cuboid? A, it's side elevation and B, it's plan view.

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

Here are answers.

Rectangle one shows a side elevation and rectangle four shows the plan view.

Rectangle three shows the front elevation if you wanted to write that down as well.

Let's look back at this example again 'cause Aisha's noticed something.

She says, "I've noticed that each length appears exactly twice.

Once in two different viewpoints.

Let's take a look.

We can see the three units appears once in the front elevation and once in side elevation.

That's twice altogether.

We can see that the four units appears twice.

Once in the front elevation and once in the plan view.

And we can see that the five units appears twice as well, once an side elevation and once in the plan view.

That means we can draw a third viewpoint if we're only given to them.

Like in this situation here we've got the front elevation and the side elevation, but we don't have the plan view, but we can work out what the plan view would look like based on the measurements we're given for the other two.

We can see that the length of five units appears twice already, but the length of two units and four units each only appear once.

So that means a plan view must be two units by four units, so four units like this and two units like this.

And our plan would look a bit like this.

Now we've got that we can find the area of each and then we can find the surface area by doing the following calculation and that would mean we'd get 76 square units for the surface area.

Let's check what we've learned.

These rectangles are the plan view and side elevation of a cuboid.

What is the area of the rectangle at the front elevation of the cuboid? Pause the video while you work it out and press play when you're ready for an answer.

Well, this is what the front elevation would look like.

You may have drawn that you may not have done.

You may have just worked out that the remaining lengths would be five units and seven units and multiplied them to get 35 square units.

Okay, it's over to you now for task B, this task contains three questions and here is question one.

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

And here are questions two and three.

Pause the video while you work through these and press play when you are ready for answers.

Okay, let's go through some answers.

Question one, you have to draw the side elevation and plan view for the cuboid and they look like what you can see on the left.

And then you need to use those to complete the calculation to find the surface area of the cuboid.

That would be two lots of 16 plus eight plus 32, and that gives you 112 square units.

Now with question two, you had the front elevation and side elevation already drawn and you had to draw the plan view and that looks like what you can see in the bottom left of the screen.

It is six squares by seven squares and then you're told that each square is one inch of length.

You had to find the surface area.

You do that and you get 214 square inches.

And question three, cuboid X has the same front elevation as the cuboid in question two, however its depth is 10 inches long and you have to find a surface area of cuboid X.

You can do that by adapting your calculations from the previous question and you'd get 310 square inches.

Well done so far.

Now let's move on to a third and final part of this lesson, which is looking at the surface area of a cuboid from isometric drawings.

Here we have another cuboid that we're going to find a surface area of, but maybe you don't want to draw the nets or don't want to draw the plan elevations in order to find a surface area.

Maybe you just want to find a surface area based on the diagram that you're given.

Well, we can find the surface area of a cuboid from its 3D representation by finding the area of the rectangle from each viewpoint.

Let's do that together with this one.

The area of the front face would be eight multiply by 16, which is 128.

The area of the side face would be 25 multiply by 16, which is 400.

Now the rectangle that you can see from the plan view doesn't have any lengths labelled on it, but it is congruent to the rectangle on the base of the cuboid.

So it would be eight centimetres by 25 centimetres.

That means its area would be 25 multiplied by eight, which is 200.

So to find the total surface area of the cuboid, we could find the sum of these areas and then multiply it by two.

And the two that doubles the sum of the three areas we've got so far, that counts for the three areas that we can't see, the three rectangles that are congruent to the ones that we can see.

So that gives us a total surface area of 1,456 centimetres squared.

Let's check what we've learned.

Here we've got a cuboid.

Which two edge lengths define the rectangle of the front view? And then what is the area of this face of the cuboid? Pause video while you work those out and press play when you're ready for answers.

The edge lengths would be 60 and 12 units, therefore the area would be the product of 60 and 12, which is 720 square units.

Here's another cuboid.

Complete the values in this calculation that starts to find the surface area of the cuboid.

In other words, find the area of the front face, find the area of the side face, and then input those numbers into the calculation you can see at the bottom in order to find the surface area.

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

The area of the front elevation will be 108 square units.

The area of the side elevation will be 240 square units, and that means your calculation will be two multiplied by the sum of 108, 240 and 180, and that would give 1056 square centimetres.

Here we have an isometric grid and cuboids can be drawn on an isometric grid rather than grid squares.

This helps draw shapes in 3D for example.

Here we've got a cuboid drawn on this isometric grid.

Let's now start thinking about the lengths on this cuboid.

And the lengths go in three different directions.

Lines going in this direction, show the width of the cuboid lines go in this vertical direction, show the height of the cuboid and lines go in this different diagonal direction.

Show the depth of the cuboid.

And then what about units? Well, we can define one full edge of a triangle as one unit of distance.

So when we look at the height of this cuboid, it is one, two, three full edges of triangles, which means it would be three units in height.

The width will be four units and the depth will be five units.

So now we have all the lengths.

We can calculate the surface area in the same way we did earlier.

We can find the area of the three faces we can see, add them together and then double it so we've got the area of six faces.

The area of the front face is 12 units squared.

Area of the side face is 15 units squared.

The plan is 20 units squared.

So the total surface area would be double the sum of those three areas, which will give 94 units squared.

Let's check what we've learned.

Here's a cuboid drawn on isometric grid.

What are the two edge lengths of the rectangle on the plan view of this cuboid? And then using that, what is the area of that face of the cuboid? Pause video while you work through it and press play when you're ready for answers.

The edge lengths are two units and six units, which means its area is 12 square units.

Okay, it's over to you now for task C, this task contains three questions and here is question one.

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

And here is question two.

Pause while you do it and press play for question three.

And here is question three.

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

Okay, let's see how we're done.

For question one, here are your edge lengths for part A.

For part B, here are your three areas.

And for part C, the total surface area will be the sum of these and then doubled, which will give you 1,266 centimetres squared.

For question two, you had two cuboids to find the surface area of and then compare.

The cuboid label A has a surface area of 4,032 square inches.

The cuboid labelled B has a surface area of 4,464 square inches, which means when we find the difference between them, we get 432 square inches.

For question three, you've got three representations of three different cuboids.

And you have to start the smallest, put the cuboids in order of size for the surface area that involves finding the surface area of each.

This cuboid represented by a net has a surface area of 536 square centimetres.

This cuboid represented by its plan and elevations has a surface area of 510 square centimetres.

And this cuboid represented by an asymmetric drawing has a surface area of 576 square centimetres.

Which means when you put 'em in order of size, the one represented by the plan elevations would be the smallest, then the one represented by a net, and then the one represented by an isometric drawing.

Fantastic work today.

Now let's summarise what we've learned.

The surface area of a cuboid can be found by calculating the area of all six rectangles of the net and then adding them together.

Or the surface area of a cuboid can be calculated more efficiently that, we could, for example, find the area of the rectangles of the front elevation, the side elevation, and plan viewpoints, and then double the sum to account for the other three faces.

Or the surface area of a cuboid can be found from an isometric drawing by first finding the edge lengths using the isometric grid.

Well done today.

Have a great day.