Lesson video

Hello, I'm Mrs. Lashley, and I'm gonna be working with you as we go through the lesson today.

I really hope you're looking forward to it and you're ready to try your best.

So our learning outcome today is to be able to calculate the volume of any prism.

On the screen, there are some key words that I'll be using during the lesson.

You've met them previously in your learning, but you may wish to pause the video here just to refamiliarize yourself before I make use of them during this lesson.

So the lesson of checking and securing understanding of volume of prisms is gonna be broken into 2 learning cycles.

So for the first learning cycle, we're gonna calculate the volume of a prism, and in the second learning cycle we're going to use the volume of a prism.

So let's make a start at looking how we calculate the volume of a prism.

A prism is a group of 3D objects that have certain properties.

So which of the 3D shapes do you think is a prism? So we've got five on the screen.

You may wish to pause the video and discuss it if you're sitting with somebody.

Which of these would we classify as a 3D shape, which we would call a prism? So B and E are the 2 of the five there that are prisms. So what makes them prisms whilst the others are not? So we're gonna have a look at that now.

So why are they prisms and the others are not? So on the left there is an example of a prism, and on the right there is a non-example of a prism.

So the cross section which is taken parallel to the base must be exactly the same as any other cross section taken parallel to the first one.

So if we look at the example that is a prism, those cross-sections, those grade hexagons are parallel to the base and they are uniform in their size.

Whereas the non-example, again hexagonal, but the top cross-section which is parallel to the base is a different size hexagon to the bottom.

So this would mean it hasn't got a uniform cross-section, which means it cannot be a prism.

The 2D cross-sectional shape must also be a polygon.

So the left hand one, the example of a prism, this prism has equal actual triangles as its cross-sectional polygon and that clearly is a polygon.

Whereas the cylinder has congruent circles and circles are not polygons.

So that is a non example of a prism.

So a cylinder has a very similar structure to a prism, but by definition it is not because it's cross-section is not a polygon.

So here is a check for you.

Jacob says, "The shape of the cross section is the same throughout this solid, so it is a prism." Is Jacob correct? So pause the video.

You may wish to go back over some of the parts of the video before you answer that question and press play when you want to check.

So although the shape of the cross-section is the same throughout, its size is not.

So because the cross-section is not uniform, then it is not a prism.

So if we think about a cuboid, which is a prism, it's a rectangular prism if you like, this cuboid in particular has a volume of 18 cubic units.

It can be thought of as 6 blocks by 3 blocks where each block is one unit by one unit by one unit.

So here are the 6 blocks, here are the 3 blocks, and when I'm talking about a block, it's one by one by one unit.

So there are 18 of those blocks in total that creates that cuboid.

If the cuboid was to be stretched by scale factor K backwards, how would the volume change? So here is a diagram of the keyboard now that is stretched by a scale factor of K backwards.

Well, the cross-sectional area is unchanged.

That rectangular face of the prism, that is a uniform cross-section, if I took parallel slices, it would still be 18 square units.

It's the depth that has changed by a factor of K.

So the volume will change, and by an increase by a factor of K as well.

Because this time we could talk about it being 6 blocks by 3 blocks, but instead of each block being one by one by one, it would now be one by one by K.

So if we focus only on prisms that have a depth of one unit in this particular point, this cuboid we've seen previously has a volume of 18 cubic units.

This triangular prism has a depth of one and it would have a volume of nine cubic units.

And this right trapezoidal prism has a volume of 15 cubic units, and we can see also has a depth of one unit.

You might be thinking, "Well, why are we limiting ourselves to a depth of one?" Well, we're gonna come onto that.

So if we look at where these values are coming from, well the area of all uniform cross-sectional face of the cuboid is 18.

The area of the uniform cross-sectional triangular face on the triangular prism is 9.

And on the right trapezoidal prism it is 15.

So the area of those faces is the same as the volume, but the dimension of the unit is different.

An area would be square units and the volume is cubic units.

Now I've got the area of the cross section and the volume for each one there.

So if you look at it, we've got the cuboid, it's area of the cross-section being a rectangle, 6 times 3 is eighteen, its volume would be 18 cubic units, 6 blocks by 3 blocks where each block is one by one by one.

On the triangular prism, 6 by 3 and then half gives you the 9, the area of that cross-sectional face.

And that would be equivalent to 9 blocks of one by one by one.

And lastly, the right trapezoidal prism, area of a trapezium, we can do by finding the sum of the parallel sides, which gives us 10, 6 and 4, times in by the perpendicular height, which is 3.

So that gives me 30 and then half and it gives me the 15 and the volume is also 15 cubic units.

So what do you notice about the area of the cross section and the volume of the prism? So pause the video there and think about that.

Talk about it if you're sat with somebody, what do you notice about the area and the volume? There was one very obvious thing to notice, which was that the area and the volume had the same value but had different units.

So within the same prism, the area of the cross section and the volume are the same numerical value but with different units, one being square units, one being cubic units.

But something else that you may or may not have noticed is that between the prisms, the proportions are the same for both area and volume.

So if we look at the cuboid, which is the first column and the triangular prism, which is the middle column, the area of the cross section of the triangular prism is half the area of the rectangular cross section for the cuboid, and the volume is half of the volume.

Why are they proportionally the same? And that's because of the depth being equal to one in both cases.

If they both had a depth of 2, the numerical value wouldn't be the same, but the proportions would be because we can cancel out one of the dimensions.

This is a 3 dimensional shape, but because one of the dimensions has got the same value, so in this case they both had a depth of one unit, then the area to area proportionality will be equal to the volume to volume proportionality.

So if we now stop having a depth of one, but stretch these by a scale factor of K backwards, so the area of the cross section hasn't changed because we are stretching in the dimension which is given us the length of the prism, then the cuboid would have a volume of 18 K cubic units.

The triangular prism would now have a volume of 9 K cubic units.

The area of the cross section has not been affected by this stretch because this stretch is in the direction of the length.

And similarly, the right trapezoidal prism would have a volume of 15 K cubic units.

So what does this show us? Well, this shows us that to get the volume of a prism, we need the area of the cross section, the uniform cross section, and then we multiply it by the length of the prism.

So here is a check.

The volume of prism A, which is on the left, is 1.

5 times the volume of prism B.

Complete the blanks, there is a missing length on the screen.

What would that value be? So pause the video and when you're ready to check, press play.

So this would be 2.

As the length of both prisms is the same, so you can see they've both got a depth of 2 centimetres, so we can think of that as a depth because of it's sort of going backwards, but we could also call it its length, the area of the cross sections will have the same proportional relationship as the volume.

So we were told that the volume between this one and a half times the volume of prism B.

So because they both had the same length, we can just consider the areas instead.

So we can calculate the area of the triangular face on prism A, 9 times 4 and divided by 2 gives us 18.

And that is one and a half times the volume of prism B.

So that will be one and a half times the area of the cross-sectional face on prism B.

Once again, only because our length is equal.

So we can take away that one dimension and just focus on the 2 dimensional element of the prism.

So for any prism, the volume is found by multiplying that cross-sectional area by its length.

And remember, depending on how it's orientated, you might think of it as a height of a prism, you may think of it as a depth of a prism, but you need 3 perpendicular lengths to get a volume, you do a product of those.

So if your area, if you identify where your cross section is, that takes up 2 of the perpendicular measures.

And so you're looking then to see which is the third one, which is perpendicular to that face.

This prism has a cross section that could be described as a composite rectilinear shape.

So the cross section is made from multiple rectangles, so the prism can be referred to as a composite rectilinear prism.

It's actually a hexagonal prism as well.

If you count how many edges that cross-section has, it's got 6 edges, therefore it's a hexagon.

It's not a regular hexagon, but it's a hexagon.

So we could describe this as a hexagonal prism, but because it has been created using multiple rectangles, then we might think of it as a composite rectilinear prism.

To find the volume, the area of the cross-section can be calculated then multiplied by the length, or the composite rectilinear prism could be split into parts and their volumes can be totaled.

It could be that we split it into 2 cuboids horizontally like this, or we can split into 2 cuboids vertically and we get these 2 cuboids.

And here is the working out of both methods.

The first method is working out the area of the cross-sectional face, which is a composite rectilinear shape, polygon, or the second method, which is where we treat them as 2 separate cuboids, we've split it, 2 separate cuboids and we're gonna total their individual volumes.

And you can see here they both come out with a volume of 51 cubic centimetres.

Again, the left hand one, the 4 times 2 plus 3 times 3 is splitting the composite area up into 2 rectangles and finding the total, and then multiplying by the length of the prism, whereas the second one is using them as 2 cuboids.

And this is an example of the distributive law which you will have met before.

So here's a check for you.

Work out the volume of this composite rectilinear prism.

Pause the video, and then when you're ready to check, press play.

I found the volume of 2 separate cuboids, and then found the total to get me my composite prism, you may have done the area of the composite face and then multiplied by the length, but our answer should agree and that's 5,832 cubic centimetres.

So onto the first task.

On question 1, you need to write the dimensions, integers only, that the cube would could have if the cube would has a volume of 72 cubic centimetres.

And for question 2, calculate the volume of each prism.

There's 3 parts there.

You are given the area of the cross section and the length or the height, depending on what the orientation is.

Press pause whilst you're working through questions 1 and 2, and when you press play, we'll move on in the task.

Question 3, I'd like you to calculate the volume of this triangular prism.

So pause the video and then when you are ready for question 4, press play.

Question 4, there are 2 parts and we can think of these as composite prisms. So part A and part B, calculate the volume.

Press pause and then when you're ready, we're gonna go through the answers to task A.

So question 1, I told you that a cuboid had a volume of 72 cubic centimetres, the task was to write down the dimensions that the keyboard could have and we were just sticking to integer only values.

So they are all on the screen.

You may wish to pause the screen so that you can check if you got them all or if you missed any.

Something to be mindful of is if you, for example, had written 3 by 2 by 12, well, that's actually the same as 2 by 3 by 12.

So don't count that as a different cuboids, it's just a cuboids that's been rotated.

Question 2, you need to calculate the volume for each of these prisms and for each of these you were given the area of the uniform cross section and then the perpendicular length to that face, which meant you just needed to define the product of the area and the length.

So the volume of the first one is 108 cubic centimetres, for B, it's 60 cubic centimetres, and for C, it's 98 cubic centimetres.

Question 3, you need to calculate the volume of this triangular prism.

So firstly, you need to find the area of the cross section, which is a triangle, and then you need to multiply it by the length.

So half times base times perpendicular height, so that's the half times 11 times 18, and then multiplied by 13, which is the length of the prism.

That's 1,287 cubic centimetres.

Do make sure you're writing your units because that will support you if you think about each stage of your calculation.

If you've done the product of 2 perpendicular measures, that will give you an area, and then when you multiply by a third perpendicular length, that's when it becomes a volume.

So question 4, calculate the volume of these prisms. As I said, you can consider these as composite prisms. On part A, you can see that the uniform cross section is a kite.

We don't have the formula for the area a kite, so we can break it into, we can split it into 2 identical triangles because of the line of symmetry.

And that's why the volume is 6 times 16 times 10 because each of those triangles has a base of 16.

It's perpendicular height, so that would be 6 and then you would half it.

So the area of one of those would be half times 6 times 16.

But we have 2 congruent triangles.

So half times 2 gives us 1, so it's just 6 times 16.

Another way you could think about this is that you could split one of the triangles to complete a rectangle, which would be 6 by 16.

So 6 by 16 is the area of the cross section, times by 10, which is the length of the prism.

So 960 cubic centimetres.

On B, so another hexagonal prism, which we can think of as a composite rectilinear prism.

And I have shown where I've split this one.

So I found the area of the cross-sectional face.

I've got a square with a rectangle, so 8 by 8 and 4 by 10.

You need to do some subtractions to work out that length of 10 but you can use the square nature of the bottom part, that bit is 8, and if the total height is 18, then that missing edge which is a vertical height would be 10 times it by 8, which is the length of our prism.

So 104 times 8 is 832 cubic centimetres.

So the second learning cycle is to make use of the volume of a prism.

So we're gonna look at if we know the volume of the prism, 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.

We've been working with that in the previous learning cycle.

But how do you find the volume of a prism? We need to get the product of 3 perpendicular lengths, which will come from having an area which will be the uniform cross section and multiplying it by the length of the prism.

But if we know the volume and the length of the prism, can we calculate the area of the cross section? Well, the answer is yes, we can use inverse operations to find the area of the cross-section.

So have a look at that.

So if we've got this prism here, and we know that it has a length of 7 metres and its volume is 301 cubic metres, what would the area of that cross-sectional face be that we've at the moment called A? Well, we can set up an equation because we know the volume is calculated by doing the area of the cross section, A, multiplied by 7, the length.

A multiplied be 7, algebraically would write as the term 7a.

So now our equation 301 equals 7a.

So our inverse operation would be to divide, and therefore A is 43.

The area of the pentagon, it's a pentagon if you count how many edges there are, is 43 square metres and you could check.

What is 43 times seven, 301, it gives us the volume.

So a check for you.

Work out the area of the L shape, given the length and volume of the prism.

So pause the video and then when you're ready to check your answer, press play.

Once again, set up the equation.

Volume equals area times length.

The area is unknown, so I've called it X, you may have used any variable.

Then if we divide by 24.

3, because that is the inverse operation to multiply by 24.

3, X comes out as 14.

So the area of the L shape or the hexagon is 14 square centimetres.

So what about if we know the area of the cross section and the volume? Can we work out the length? So here I've got another L shape prism or a hexagonal prism.

The area of that uniform cross-section is 20 square centimetres, and the volume of the prism is 268 cubic centimetres.

So how do we find the length? Once again set up an equation.

Volume is given by doing area multiplied by length.

I've just simplified that to 20 L, divide by 20 because that would be the inverse operation, and so the length of the prism is 13.

4 centimetres.

So another check for you.

Work out the height of the prism given the area of the L shape and the volume.

So pause the video, set up your equation, solve it, and when you're ready to check your answer, press play.

So the height of the prism would be 12.

4 centimetres.

So note here that we're calling our height, and that is because of the way that the prism has been orientated.

So there is sort of sitting on one of those cross-sectional faces.

So that perpendicular length is going upwards, we call that a height.

So if we have a look at here, we've got a triangular prism, we've got a perpendicular height of 6 centimetres, an unknown edge length of the triangle X, and the length is 7, the volume is 210 cubic centimetres.

So we're gonna still set up our equation.

The difference here is we are not just wanting to find out the area of the cross section, we want to find a length that has contributed to the area of the cross section.

So it's a triangle, half times base times perpendicular height.

So we've placed those in, we've substituted that in, times by the length, which is 7, the half times X times 6 does simplify to 3x, 3x multiplied by 7 or 7 lots of 3x is 21x.

You can probably work out very quickly in your head the X would be 10.

So the base of the triangle is 10 centimetres.

A check like that for you.

Work out the length of Y, Y is the perpendicular height of the triangle.

So pause the video and when you're ready to check your answer, press play.

So Y is equal to 6 centimetres Once again, you're gonna set up your equation.

Volume equals area of the cross section, multiplied by the length.

What we don't have is one of the dimensions of the triangle.

So we are gonna put a variable in there, which I chose as Y.

We end up with seven halves Y, you may have written that as 3.

5y.

Believing it as an improper fraction is what we would encourage.

It does simplify to 42y and then when you divide by 42, you get 6.

This is not the only steps to solve.

You may have divided by 12 straight away and then divided by 7 and then divided by a half and you'd still get Y equals to 6.

So one to the last task of the lesson where question 1, you need to work out the marked length or area that each prism.

So there's 3 parts to this question.

Press pause whilst you work out the answers to A, B, and C.

When you're ready to move on with the task, press play.

Here we have question 2.

So you need to work out the marked length for each prism.

So there's part A and part B.

So on A, it's X that you're trying to work out, the length, and Y on part B.

Pause the video and then when you finish your question 2 and you press play, we're gonna go through our answers to both of these questions.

Question 1, A was 7.

So on part A, you were told the volume and you told the area of the cross section, so you need to do 91 divided by 30 to give you that length.

On B, you were given the length of the prism and the volume of the prism, and you wanted the area, so you just needed to do 294 divided by 14.

And so B equals 21.

And on C, you were given the area of the cross section, the volume, and you needed the height of this prism.

So 144 divided by 9 gives you 16.

So A equals 7, B equals 21, and C equals 16.

And then this last question, I'm gonna do part A and then we'll move to part B.

So work out the march length for each prism.

So this was a triangular prism, and X was the perpendicular height of the triangle.

You were given 6 centimetres as the base of the triangle and the height of the prism was 14.

The total volume was 126 cubic centimetres.

So I've set up the equation.

I've got the area of my triangle, half times base times the unknown perpendicular height X.

Then that is gonna multiply with the height of the prism to give me the volume of 126.

I've simplified my area, got to the term 3x, then I've simplified that algebraic expression, 3x times 14 is 42x, and then use the inverse operations to divide 126 by 42.

So therefore X is 3.

And on to B, this was a trapezoidal prism, and 15 and 20 are the parallel edges of the trapezium.

We know they're parallel from the 2 right angles that this trapezium has.

The length of the prism is 8 metres and the volume is 840 cubic metres.

So again, set up the equation, the area of the cross section.

Well, how do you work out the area of any trapezium? You need to do the sum of the parallel sides, multiply it by the perpendicular length and half it.

So a half times 20 plus 15 times Y.

Then that would be the area of the cross section.

For a prisms volume, you need to multiply it by its length, which in this case is 8 and we know the volume is 840.

The way that I've simplified this is I've applied the half to the 8.

Because this is a product, I can do that in any order.

So I've applied the half to the 8, which is why there is a 4, and 20 plus 15 gives me 35, 35 times y is 35y.

I've then divided both sides by 4, which is where the next line of working 35y equals 210 comes from, and then divided both sides by 35 using the inverse operations, and Y equals 6.

So the perpendicular height of the trapezium is 6 metres.

So to summarise today's lesson, which was checking and securing understanding of volume of prisms, the volume of any prism is the product of the area of the cross-sectional face and the length of the prism.

You can use this relationship to calculate the volume if you know the area and length, or if you have enough information to find the area, or if you're given the volume, you can calculate unknown areas or lengths.

Really well done today, and I look forward to working with you again in the future.