Lesson video

Hello, I'm Mrs. Lashley and I'm really looking forward to working with you as we go through this lesson today.

So by the end of today's lesson, we're hoping that we can calculate the volume of prisms using the constant cross-sectional area properties.

During the lesson, there will be some words that you have met before and their definitions are currently on the screen.

You may want to pause the video so that you can remind yourself about those before we move on.

In the first learning cycle, we're going to calculate the volume of any prism, and then in the second learning cycle, we're going to use the volume of any prism instead.

So we're going to make a start now on that first learning cycle, thinking about calculating the volume.

Aisha and Izzy have been given 24 cubes by their teacher.

Their teacher has asked them to use all of the cubes or blocks to create a cuboid.

Aisha and Izzy have started by putting six together and then another six on top, another six.

And then all of them have now been used to make this cuboid.

Izzy wonders if that was the only way that they could have put the 24 blocks together to make a cuboid because she thinks there is another way and that she can see it.

Let's have a look at that.

So Izzy had sort of noticed that her cuboid could be sort of sliced in half and that half just placed in a different position.

All 24 cubes have still been used to create this cuboid.

Aisha has also seen a different way of doing it.

Again, Aisha saw it as a sort of slice, well, two slices and has now rearranged them to make this cuboid.

Are there any other ways that the 24 blocks could be put together? You may want to pause the video to consider that.

Maybe sketch them out on a piece of paper before I show them.

So this is another way.

This way.

So just a sort of a long thin 24 cubes in a line.

Or this way.

So here they all are on one slide.

The one they originally did, the one that Izzy noticed, the one that Aisha noticed, and then the alternative ones.

How well did you get on? Quick check.

Which of these cuboids are made from 18 cubes? So instead of 24 like Aisha and Izzy had, if you only had 18 cubes, which of these would you be able to create? Pause the video and then when you're ready to check your answer, press play.

So you would've been able to do the second and the third one on the screen.

They are both created from 18 cubes.

How many cubes have been used for this cuboid on the screen? Well, Laura knows that there's 48 and that's because she sees it as 4 layers of 12 cubes.

And we can sort of see that sort of exaggerated diagram to the right whether we've got, we can see those four layers that are all identical to each other, each made of 12 cubes.

Andeep agrees with Laura.

He also thinks there are 48 cubes, but he doesn't see it as layers.

But instead sort of like slices and this time slices of 16 cubes.

So here we've got a cuboid, but instead of seeing the individual cubes, we've just been told how many in each direction in the three-dimensional directions.

So there's three cubes, that's sort of the width of the cuboid, five cubes, we could think of as the length of the cuboid, and two cubes is the height.

So how many cubes would've made this cuboid? So you could think of it as 2 layers of 15 cubes and that sort of darker purple is the top face of the cuboid.

And because it would be 3 cubes wide and 5 cubes long, that would be 15.

We can see that there are 2 slices of 15 cubes.

But alternatively, you might think of it as 3 slices of 10 cubes.

And now the face that we're getting the 10 from is this to the right, that purple face, and it's 5 long and 2 high and so that would be 10 cubes, and we can think of it as 3 slices.

Alternatively, you might be looking at it as 5 slices of 6 cubes.

And the 6 is coming from the other purple face, which is 3 cubes wide and 2 cubes high.

Alex has collected all of those together.

2 lots of 15 cubes when we thought of it like layers, 3 lots of 10 or 5 lots of 6, they all result to the same thing, 30 cubes.

They are all factor pairs of 30.

So what about this one? If it's 3 cubes wide, 5 cubes long but 3 cubes high.

Well, one slice would be 9 cubes and there would be 5 of those if we're thinking of the slices from that face.

And so 9 lots of 5 or 5 lots of 9 really because there's 9 cubes in each slice and there are 5 slices, would be a total of 45 cubes.

So 45 cubes would've created this cuboid.

If we thought of it as like a layers problem instead, then one layer would be 15 cubes and there would be 3 layers up the cuboid, and so 45.

So here is a check for you.

How many cubes would make this cuboid? So think about it like slices or think of it like layers, but how many cubes would you need to create this cuboid with these dimensions? Pause the video whilst you calculate that and when you're ready to come and check, press play and then we'll move on.

216 cubes would be needed to create a cuboid with these cube dimensions.

108 by 2, and that's where you're thinking it was 2 sort of slices or 24 times 9.

And that again would be 9 layers or 9 slices or 18 times 12.

And that one's really the layers one where 18 is the base face, 2 times 9, and then moving up the cuboid, all of their product is 216.

So what is volume? This lesson is all about finding the volume of any prism, but what actually is volume? So volume is the amount of 3D space that is taken up by a shape.

Finding the volume of any shape, any 3D solid, always involves multiplying together three perpendicular lengths somewhere within the calculation.

So if we've got our cuboid that we've currently been working with, the three dimensions and the three different perpendicular lengths, well, we've got, when we've been thinking about these slices, well, first of all, we are just sort of working out the area of a face and there's two perpendicular lengths to get the area and then we multiply it by the third length, which is perpendicular to that face and that becomes our three dimensions, which gives us our volume.

So volume is a measure in cubic units or cubed units because you've multiplied a length by a length by a length, you've cubed a length.

Examples for volume units, kilometre cubed or cubic kilometres, cubic metres, cubic centimetres, cubic millimetres, they're all of our metric units or some of our metric units I should say.

And then we've got some imperial measurements as well.

So cubic miles, cubic feet and cubic inches.

When you go to a builder's yard, you might buy a quantity of sand or a quantity of concrete and they might talk to you about how much do you want in cubic metres or cubic feet or if you're thinking about a swimming pool and you need to fill it up with water, then there might be, again, the cubic capacity of water.

So you'll hear this and that's about volume and that's to do with the shape and the amount of space taken up by a shape.

So if each of these cubes that we were building the cuboids from all the way back to the start when they were given the cubes from the teacher, if each one of those has a volume, an individual volume of one cubic centimetre, then what would the volume of the cuboid made from them be? How much space is this cuboid taking up? So a single layer, we had already spoken about would be 15 cubes because we can think of it as that on the base that 3 cubes by 5 cubes would be 15 cubes.

So that would have a volume of 15 cubic centimetres because each one of them has a volume of one.

And we've got 3 layers of those 15 cubes or of those 15 cubic centimetres, which means that overall, the cuboid would have a volume of 45 cubic centimetres.

Each layer has got 15 cubic centimetres and there are 3 layers in this cuboid and therefore 45 cubic centimetres.

The volume of this cuboid is fixed at 45 cubic centimetres regardless of what size cubes you were filling the cuboid with.

So we just made one of those blocks have a one cubic centimetre volume, but if you had even smaller ones, then their volume would be smaller and some more cubes would fit within it.

But the total would still be 45 cubic centimetres or would be equivalent to 45 cubic centimetres.

And we can think of filling it up with water.

I already said that we'd speak about the capacity of a swimming pool in terms of cubic metres or cubic centimetres.

So if you imagine filling it up with water, what would happen? Well, the base would fill out first, the water would cover the base, and then as more liquid or water start to fill it, then layers would start as it rises up in the cuboid and it would continue until it is completely full, however many layers that might be because of the capacity and the volume of each drop of liquid.

So Jun and Sofia are discussing how to calculate the volume for any prism.

So we've got this concept that volume is how much space a 3D shape takes up.

And depending on if we think about building from blocks, then each block would have its own volume and we could calculate a volume from using how many blocks we've taken.

But blocks could be of various sizes.

And so it's not always just if you've used that many blocks, that is the volume, it depends on the volume of each block.

And also we could fill the space up using a liquid instead of an individual cube.

So Jun said a prism has a uniform cross-section throughout the length of it.

And that's part of the definition.

It's not the full definition of a prism, but a prism does need to have a uniform cross-section.

We call that the base that runs through the length of the prism.

And Sofia agrees and says, "Yes, a cuboid has a rectangular cross-section." So if you think about a cuboid, we see them a lot in your house, in your school, that you might have a tissue box, that's a cuboid, or a cereal box which is a cuboid and each of the faces are rectangular and actually depending on which way you want to look at it, each of those faces could be the cross-section.

So a rectangular cross-section for a cuboid.

Jun says to find the volume of a cuboid, you find the area of the rectangle, so that's the cross-section, and then you multiply it by the height.

Sofia says if it was a triangular prism, it is layers of triangles.

So we think about these layers that we spoke about before, slices or layers that we could see the rectangles which were actually cuboids themselves 'cause they were one block deep each layer building up to build our prism.

So to find the volume of a triangular prism, find the area of the triangle and multiply it by the perpendicular distance between the bases.

Jun's asking that as a question to Sofia, he's not a hundred percent sure, but you can see that she just said a triangular prism is like layers of triangles stacked on top of each other.

So his suggestion, well, maybe is the volume then the area of one of the triangles multiplied by the perpendicular distance between the two congruent bases of a prism? Sophia says yes.

She's reassuring him, yes, that's exactly how we do it, but sometimes that perpendicular distance can be called different things.

It could be a height, it could be a length depending on how it's presented.

So if the prism is sort of sat on its base, then you're going to think of the prism having a height because it sort of comes up from the surface it's sat on.

However, if it was a right prism, if we're laying on one of the rectangular faces, then we'd probably talk about the prism having a length.

But when it comes to the calculation for the volume, it's the perpendicular distance and remembering that volume is at some point in the calculation, three perpendicular lengths have been multiplied together.

So the area would be two of those perpendicular lengths and then you multiply it by the height or the length to give you your third and your cubed unit.

We'll take a look at this example here, which is a triangular prism.

It's a right triangular prism and it's a right-angled triangular prism.

So Jun said we find the area of the triangle, which is the base or the uniform cross-section of the prism.

And so in this case, it is 3 centimetres by 4 centimetres and then divided by 2 because it's a triangle.

So half times base times perpendicular height, which gives you an area of 6 square centimetres.

And then that height of the prism is 8 centimetres.

So if we think of this as layers, then because it is 8 centimetres, we could think of it as 8 one-centimeter layers and each of them would have a volume of 6 cubic centimetres because 1 centimetre multiplied by 6 square centimetres gives you 6 cubic centimetres.

So each layer has a volume of 6 cubic centimetres and there would be 8 one-centimeter deep layers.

So overall, the volume would be 48 cubic centimetres.

What did we actually just do? Well, we found the area of the cross-section and we multiplied it by the perpendicular distance between the parallel bases.

In this case, it was a height because of the orientation of the prism.

Jun wants to extend this for any prism, not just a cuboid, not just a triangular prism but any prism.

So the volume will be the area of the cross-sectional face, whatever shape that might be, multiplied by the length.

And we can see here that Sofia has said, yes, prisms have got this constant cross-section.

It's not the only part of the definition but it's a key part of the definition.

If a 3D shape is to be a prism, it does need to have a uniform cross-section.

And then you multiply it by the length and that's like thinking about how many one centimetre deep layers there are.

This works even if there aren't a whole number of one unit deep layers.

So a quick check for you, what would the volume of this pentagonal prism be? The area of the pentagon is 12 square centimetres and its height is 8 centimetres.

Pause the video whilst you calculate the volume and remember your units as well, and then press play to check.

Jun just told us that to find the volume for any prism, so this one's a pentagonal prism, area of the cross-section, which was given to you in this check, 12 centimetres squared, and then you need to multiply it by the length.

So 12 times 8 is 96 and the units would be cubic centimetres.

It is a volume and so the units need to be cubed.

On the screen, there is a diagram of a swimming pool and the swimming pool has got a trapezium cross-section.

So you've got the feathers there to indicate the parallel sides.

We know that a trapezium has at least one set of parallel sides and we need that to work out the area.

So if swimming pool is 18 metres wide and 50 metres long and has got this trapezium cross-section, and if you've ever been to a swimming pool, we tend to have a shallow end and a deep end and the floor of the swimming pool is sort of tapering down and that's what the trapezium is doing.

So what is the volume of the pool? First of all, for any prism, we need to get the area of the cross-section.

So because it is told to us to be a trapezium, we need to recall the area of a trapezium formula, which is half times the sum of the parallel sides multiplied by the perpendicular height between them or length between them.

So the half is written as a decimal here, 0.

5, the parallel sides we know from the feather marks on the diagram, so 1.

2 and 1.

8, and then multiplied by the perpendicular distance between them, which is the length of the swimming pool, in this context, of 50.

So the area of the cross-sectional face is 75 square metres.

We want the volume though, so we now need to multiply that area of the cross-section by the perpendicular distance or the width of the swimming pool.

So 75 multiplied by 18 gives you 1,350 cubic metres.

You are now going to do some work independently thinking about volumes of any prism.

So question one, you need to sketch all the cuboids with integer lengths that have a volume of 18 cubic centimetres.

Question two, you need to calculate the volume of each of the three cuboids that you can see on the screen.

Be careful of any mixed units, you need them consistent before you start doing any calculations.

Pause the video whilst you make a start on that and then when you're ready to move on, press play.

Question three, there are pairs of shapes with the same volume.

So as you can see on the screen, there are five prisms on the screen.

So one of them does not have the pair.

So which of those shapes on the screen doesn't have a pair? They are paired up by their volumes.

If they have the same volume, then they are a pair.

So pause the video whilst you're trying to figure out which one is the odd one out that doesn't have a pairing.

Press play when you're ready for question four.

Question four has got quite a complicated-looking diagram.

The arrows are there to indicate which lengths they're speaking about.

There is mixed units as well, so be careful it's not drawn to scale.

So it's a diagram of a mould for a structural beam.

Steel is poured into it to create the beam.

So it's got an opening at the top where they're going to pour the steel in and it will form this sort of I-shaped prism, which will be a structural beam in a building.

How much steel can it hold? So pause the video whilst you're calculating how much steel this mould for the steel beam can hold.

Press play and then we'll go through the answers to questions one, two, three, and four.

So question one, you need to sketch all the cuboids with integer lengths that have a volume of 18 cubic centimetres.

I've not got the sketches on here, but I've got the dimensions.

So there were four that you could have come up with.

There were four that you could have chosen with integer lengths.

There would be many more that you could have come up with with decimals and you can challenge yourself to do that.

But for integer only, there was just the four options.

Question two, you needed to work out the volume of each of the cuboids on the screen.

Question part a, they were all in centimetres, there wasn't any issues with consistency of units.

You needed to do 3 times 2 to get the area of the cross-section and then multiply it by 4, which was the length in this case.

You may have done 3 by 4 to get 12 and then multiplied it by 2 and thinking of it like layers or the height of the cuboid and you'd get the same answer because of the commutative nature of multiplication.

Cubic centimetres was your unit.

On b, there was mixed units so you needed to be careful here.

So 50 millimetres, convert that to 5 centimetres and then you can calculate the volume of the cuboid and it would be 10 cubic centimetres.

You may have converted the centimetres into millimetres.

The reason I've not done that on the screen is because that takes two conversions.

There was one that was the odd one if you like.

So it was more efficient to change that one than to change the other two.

If it stipulated that it needed it in cubic millimetres, then of course you would've had to have changed your centimetres into millimetres.

Part c, we've got some decimal calculation here, 40.

96 cubic metres.

Question three.

So there were pairs which had the same volume and then there was the odd one out, the one that didn't have a pairing and the one that didn't have the pairing was the hexagonal prism.

Two of them had a volume of 14, two of them had a volume of 9, and then that hexagonal prism had a volume of 12, so it didn't have a pair.

Question four, it was being poured into it to create the beam.

So how much steel is how much space it takes up.

It is a volume question.

So you need to do the area of the cross-sectional face.

It was a composite shape, the I shape.

So you may have split it as many ways you may have split it.

The way that I've split it and written the calculation on the screen is two rectangles, the sort of two horizontal parts of the I, and then the third rectangle, that central part.

The dimensions for that part were given in centimetres so I've converted them into metres first to calculate my area so that I was working in metres.

And then I've multiplied it by the length of the prism, which gives the volume as 0.

2688 cubic metres.

We're now onto the second part of the lesson and the second part is going to make use of the volume.

If we already know the volume, what else can we work out? So if we know the area of the cross-section and the length, then we can calculate the volume.

That's what we were just doing in the first learning cycle when you've been doing in your task.

Area of cross-section multiplied by the length is the way to find the volume for any prism.

If we know the volume and the length of the prism, can we calculate the area of the cross-section? So, we can and we can make use of our inverse operations to find the area of the cross-section because the volume was calculated by doing a multiplication.

So we can use the inverse of multiplication division to work out the area of the cross-section.

So let's have a look at one of those questions.

Here we have a irregular pentagonal prism and we've been told that it has an area of a square metres, don't know the value of a, we need to find that out.

What we do know is that the volume of this prism is 344 cubic metres and its length is 8 metres.

So if we substitute what we do know into our sort of formula, we end up with this equation, 344 equals a, that's the area of the cross-section that we don't know, multiplied by 8, which is the length of this prism.

a to multiply by 8 algebraically, we should write that as a term 8a.

So 344 equals 8a.

And we are trying to solve this to find a.

We're going to do that by dividing by 8, which leaves us with 43 equals a.

Because it's an equation, each side is balanced, we could write that as a equals 43.

And what was a in this question? Well, a was the area of the pentagon, so I can write that out as the area of the pentagon is 43 square metres.

It's not cubic metres anymore because it's not the volume, we have divided by a perpendicular length so the units have decreased by one dimension.

It's gone from a cube to unit down to a squared unit.

Here's another one.

This time we've got an L-shaped prism or an irregular hexagonal prism.

Again, we don't know the area of that face, we've just called it b.

The length is 13.

5 and the volume is 324.

So substituting into the formula for volume of any prism, we get 324 equals b times 13.

5.

Again simplifying that as a algebraic term removing the multiplication symbol, we get 13.

5b.

You could write that as an improper fraction 27/2b, that would be equivalent to avoid the decimal within this equation.

Once again, we're going to use our inverse operations to solve for b.

And b is 24.

So the area of the irregular hexagon is 24 square centimetres.

Here's a check for you to do following on from that.

So this is another L-shape prism and given its length and its volume, what is the area of the cross-section of the L-shape? Pause the video whilst you calculate that and then when you're ready to check it, press play and we'll carry on.

So substituting it in, simplifying it, dividing, you hopefully got to 12.

5.

So the area of the L shape is 12.

5 or twelve and a half square centimetres.

Well done if you've got that one right.

So we've just seen that you can use the volume and the length to get the area of the cross-section.

So what about if we know the area of the cross-section and we know the volume, can we get the length? So we're going to substitute in, we know what the volume is, we know what the area of the cross-section is and it's the length that we don't know.

We can write that as 24l instead so we don't have our multiplication symbol.

Using division, we get 8.

5.

The length of this prism would be 8.

5 centimetres.

So one for you to do.

So work out the height, remember that this is the height because of the way that the prism has been sat, but no different to length.

So given the area, the area is 52 and given the volume, 379.

6, what is the height? Pause the video whilst you calculate your height and then press play to check it.

Putting it into the formula for volume.

We know the volume, we know the area.

What we don't know is the height of the prism.

Simplifying up that term to 52h.

Using our inverse operations, height would be 7.

3 and the units are centimetres.

So the height of the prism is 7.

3 centimetres given that volume and that area.

So now we've got a prism, a triangular prism where it's not the area of the cross-section, the triangle, or the length of the prism that we're trying to find, instead it is actually an edge length of the triangle.

We know the volume, it's 84 cubic centimetres.

So using that formula, we've still substituted what we know, but this time we know that it's a triangular prism.

And so the area of the cross-section is actually the area of a triangle.

So 84 equals 1/2 multiplied by the base and it's the base that we don't know, so we've put the x in there, multiplied by the perpendicular height, which is 4, that's shown in the diagram, and then multiplied by the length of the prism, which is 7.

So we've now got to simplify the right-hand side of the equation before we can start to solve it.

So we can simplify the part in the brackets.

The part in the brackets is the area of the triangle.

So 1/2 times the 4 gives you the 2x and then we've got multiplied by the 7 and that simplifies to 14x.

It might be that you don't need the intermediate line there, that you can do 1/2 times x times 4 times 7 and simplify that to 14x in one go and that's absolutely fine.

So now we need to use the inverse operation of division to work out the value of x, and x would be 6.

So we've now got that the base of the triangle on this triangular prism is 6 centimetres.

So similar check for you, but you are this time working out the perpendicular height of the triangle.

So pause the video whilst you're working out the perpendicular height, which has been labelled as y, and then when you're ready to check, press play.

So substituting it all into the formula, simplifying, again, you may not have as many lines of working, but 135 equals 27y and therefore y is 5.

So the perpendicular height of the triangle on this triangular prism is 5 centimetres.

Now we've got two cartons, cuboid cartons which is what you tend to get.

So these two cartons hold the same amount of liquid.

How tall is carton A? So here we've got a problem where it's about how much liquid they hold and we've seen with the swimming pool and the concept of volume that how much space it takes up and how much liquid it holds are sort of connected.

So it is a volume question.

We need to work out how tall carton A is or the height of A.

What we have been told though is that they hold the same amount of liquid, which means they have the same volume and carton B has got all of the dimensions on it, which means we can calculate the volume of B.

And if that volume of B is equal to the volume of A, it helps us get to the height of carton A.

So let's have a look at that.

Volume of carton A equals the volume of carton B.

We know that because of the statement that says they hold the same amount of liquid.

So on the left-hand side, we know that the calculation for volume, for volume of A, would be 4 times 5, that would be the area of the cross-section, the base, and then multiplied by the height, which we don't know, but we can write in the x.

On the right-hand side, it would be the volume of B, which is the cross-section, 7 times 5, and then multiplied by the height, which we do have, which is 6.

4.

We can simplify everything we can there.

So 4 times 5 we know is 20, we can evaluate that.

And the right-hand side, 7 times 5 times 6.

4 is 224.

Now we've got an equation that says 20 times something.

The height of carton A is equal to 224.

We can use our inverse to work out, the x would be 11.

2 centimetres.

You are on the last task section of the lesson.

So the first task is, here is a right trapezoidal prism and you need to work out the missing parts of the calculation all the way down to work out the perpendicular height x.

Pause the video whilst you're working through that calculation and then when you're ready for the next question, press play.

So question two says that both prisms have a volume of 378 cubic millimetres and you need to work out the missing length in each of the prisms. So there's a right-angle triangular prism on the left and there is a prism with a base that is a kite on the right.

So both of them have got a volume of 378 cubic millimetres.

Pause the video whilst you're working out the answers and figuring out what A is and what B is, and then when you're ready for the next question, press play.

So last question of the task and last question of this lesson.

Prism A and prism B have the same volume.

Work out the missing length on prism B.

So on the left, we've got a trapezoid or prism, and on the right we've got an L-shaped prism.

They have the same volume and that's where you're going to get started in this problem.

Pause the video whilst you're working it out and then when you're ready, press play and we're going to go through the answers to questions one, two, and three.

So question one, you need to complete the missing parts of the calculation.

So it was a trapezoidal prism.

So we have 1/2 times the sum of the parallel sides multiplied by the length between them and then you need to multiply it by the length of the prism to get the volume.

And we know the volume is 300.

The missing value was the other parallel side, so 7.

5.

Then on the second line, we were just simplifying that sum, the sum of the parallel sides.

So the missing value would be 20.

Then that simplifies down to 10x.

On the right-hand side, it would be 60.

And that's because we've done the division by 5 as well.

So the 10x has come from doing 1/2 times 20 to give you 10, 10x, and multiplication by 5 is no longer on the left-hand side.

So that means we've used inverse operations to remove it, to make it one, so 300 divided by 5 gives you 60 and therefore, x, that missing length is 6, 6 metres in this case.

Question two, there was two prisms. We're going to go through the first one here.

We know the volume was 378.

You've told that in the words of the question.

It was a right-angle triangular prism.

So the area of the cross-section is 1/2 times base times perpendicular height, and the length or the height of the prism was 7 and it equals the volume.

So solving it, you should have got 9 millimetres for A.

On the kite, the prism with the base, that is a kite.

We've got 21 times 6 times b, b being the height of the prism, and that equals that volume of 378 and therefore b is 3 millimetres.

21 times 6 worked to get the area of the kite because it's two congruent triangles and therefore you'd have done 1/2 times base times height and then you would've doubled it.

And so half and doubling would've become one and basically base times perpendicular height.

Question three, you were told they have the same volume and you needed to get the missing length on B, which was marked as an x.

So volume of A equals the volume of B because they have the same volume.

The first prism was a trapezoidal prism.

So again, the area of the cross-section for a trapezium, 1/2 times the sum of the parallel sides times the height between them, then multiplying it by the length of the prism and that equals on the right-hand side was the area of the cross-section, which was a composite shape, an L shape, so two rectangles sum together multiplied by the length of the prism, which we didn't know.

Simplifying it all down, you can solve it to find the x is 7.

So in summary, the volume of any prism, regardless of what that shape of the cross-section is, will be the product.

So to multiply the area of the cross-sectional face and the length of the prism.

So we can use this relationship between the area of the cross-sectional face, the length and volume, to work out missing lengths on a diagram or within a prism.

Well done for working through today's lesson and I look forward to working again with you in the future.