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.

Our outcome today is to be able to calculate the volume of a pyramid.

This screen has got various keywords that I'll be using during the lesson.

They are words that you should be familiar with, but you may wish to pause the video here and read them just to make sure you're comfortable and familiar before we move on with the lesson.

So our lesson on finding the volume of a pyramid is broken into two learning cycles.

The first is calculating the volume of a pyramid, and the second learning cycle will be using the volume of a pyramid.

So let's make a start at looking how do we calculate the volume of a pyramid? So, a pyramid is a 3D shape that has a polygonal base and triangular faces that meet at an apex.

And we can see examples of pyramids and non-examples of pyramids on the screen.

So on the left, they are pyramids, they've got polygonal base, and all the triangular faces meet at an apex.

On the right-hand side, the non-examples, we've got a cone, which is not a pyramid because its base is circular.

A circle is not a polygon.

Then we've got a triangular prism.

It has triangular faces, but they don't meet at an apex.

And then lastly, we've got a wonderful looking 3D solid, but it has too many non-triangular faces, because if you look at those pyramids, that, at most, you only have one non-triangular face.

So here's a quick check for you.

This is a net of a pyramid, true or false? Pause the video and then when you're ready to go onto the justification, press Play.

So, true.

Now justify.

Is it because all the faces are polygonal? Or is it because the base is hexagonal and there are six triangular faces that meets at an apex? Pause the video and when you're ready to check your justification, press Play.

It's because of the base being hexagonal and there being six triangular faces that would meet at an apex.

So this is a square-based oblique pyramid, where the apex is directly above one of the vertices of the square.

So it's a oblique pyramid because the apex is not above the centre point of the base.

Its apex, in this case, is actually above one of the vertices.

The perpendicular height of this pyramid is equal to the length of the square.

So if the square is a by a, the perpendicular height is also a.

Andeep comments to say, "That looks like part of a cube." Do you agree with Andeep? So, a cube has got perpendicular lengths that are all equal, so a by a by a, and that bottom-right vertex, they're all perpendicular faces.

So that's similar to the corner, if you like, of a cube.

Andeep has actually been told that three of these particular pyramids will form a cube.

He recognises that the pyramids have two right-angled isosceles triangular faces, which will help make squares when combined with another one.

He knows that they are right-angled isosceles triangular faces because they've come off the corner where there was three perpendicular faces and that makes it the right-angle part, and it would be isosceles because the dimensions were the same.

They were a in both direction.

So then three do form a cube.

The volume of this cube made from those three pyramids would be a cubed cubic unit because it's a by a by a.

Andeep says, "Well, three congruent pyramids made this cube.

So the volume of each one is a third of the volume of the cube." So the volume for one of those pyramids where the apex was above the vertex of a square, and the perpendicular height was equal to the length of the square is 1/3 a cubed units cubed.

So it's a cubic units because it's a volume and it's 1/3 of the volume of the cube.

So if the pyramid's perpendicular height is different to the base lengths, such as h, then this is a stretch in the vertical direction.

I'm calling it the vertical direction because we've sort of got the pyramid sat on its base.

So I'm stretching it upwards vertically.

So this time it's not a cubed.

So the volume of this pyramid needs to be multiplied by the scale factor.

The scale factor will be h over a, because how did we get from a to h? By using the scale factor h over a.

So the volume of the pyramid will be multiplied by h over a this scale factor.

It is only a linear scale factor because I'm only stretching in one of the three directions.

I'm not stretching in multiple directions.

I'm only stretching in one direction.

So it's only changing one of the lengths, which is why I'm multiplying by a linear scale factor.

So h over a multiplied by a third a cubed will become a third a squared h.

So for any square-based pyramid where the apex is directly above one of the vertices of the base, we have a formula to find the volume.

So for any of these types of pyramids, we know that it's a third times a squared, which is the area of the base, times by h, which is this perpendicular height.

But what about pyramids where this is not the case? For example, a right pyramid, where the apex is central to the base.

Let's have a look at this because this is a very particular pyramid.

Most pyramids aren't gonna look like this.

So if we take that pyramid and slice it horizontally, parallel to the base, the volume would be unchanged.

We're gonna go from having one solid to having, in this case, four solids.

But the volume, the combined volume, is unchanged.

Now, if we think about sliding the slices across to try and move the apex to the middle of the base, it doesn't look like a pyramid anymore.

Those slices is a pyramid on the top, that very top slice is a pyramid, but those other ones are not pyramids, but it will have the same base area.

The purple one at the bottom is the same base that we started with.

It will have the same volume because all we've done is sliced it up.

We haven't removed any of it or added any to it.

And it also will have the same perpendicular height.

So that distance from the base to the apex will be unchanged.

This time we're gonna increase the number of slices.

So we've now got 10 slices, and this will still have the same volume.

I've not removed any volume, I've not increased it.

It's still got the same base area, and it's still got the perpendicular height.

There's still some steps.

This is still not a pyramid, but what if we had an infinite number of slices? Imagine we had an infinite number of slices and we slid them across to get the apex in the middle.

Well, with an infinite number of slices, the edges would be smooth.

There wouldn't be this step that you can see.

More importantly, the pyramid with the same base area and the same perpendicular height and the same volume.

So despite the fact that the apex is no longer above the vertex of the base, it's now central, the area of the base would be the same, the perpendicular height of the pyramid would be the same, and its volume would be the same.

So this establishes that for any square-based pyramid, volume can be calculated using this formula.

A third times a squared times h.

Where a squared does the area of the square, the base of the pyramid, and h is this perpendicular height, regardless of where the apex is located.

But what if we stretch the pyramid in one direction and so that the square base becomes a rectangular base? So instead of being a by a, it's now a by b.

Well, the volume will be increased by a scale factor b over a.

Once again, it's a linear scale factor because I'm only increasing in one of the three directions.

So the volume would be b over a times a third a squared h.

That formula we have for square-based pyramids.

And this gives us a third abh.

Well, a third abh is ab, which is the rectangle's area and the h being the perpendicular height.

So if this rectangular-based pyramid was split into two triangular-based pyramids, like this, then each of those triangular-based pyramids would be half of the volume from the rectangular-based pyramid.

I've just cut it in half.

So a half of a third times abh is 1/6 abh.

Hmm, that looks different to the other ones that we've been working with, but we can rewrite that as 1/3 times half ab times h.

What do you notice? Well, I'm hoping you notice that that third is still there, that constant of a third.

Half ab is the area of a triangle with a base and a perpendicular height of a.

And the h is the perpendicular height of the pyramid.

So we've still got this kind of form of third times area of base times perpendicular height.

That triangular-based pyramid, which we have a formula for the volume of, has the same dimensions as this right-angle triangular prism.

This right-angle triangular prism is base times a, and a perpendicular height with the prism's length is h.

And how do we get the volume of a prism? Well, you do the area of the cross section, which in this case is the triangle, times by its length, but in this case it's a height.

What do you notice about that? So the prism is three times the volume of the pyramid.

The volume of the pyramid was 1/3 times a half ab times height.

And the volume of this prism is a half ab times height.

So a prism with the same dimensions as the pyramid is three times the size.

Well, we saw that before.

We saw that with the cube.

The cube was a cubed and three pyramids made that cube, and so it was a third a cubed.

So here's a check.

A pentagonal prism has the same height as a pentagonal-based pyramid.

The pentagonal faces in both shapes are congruent.

If the pentagonal-based pyramid is filled with water and poured into the pentagonal prism, what fraction of the prism will have been filled? Pause the video, and when you're ready to check your answer, press Play.

Okay.

It would be a third of the prism.

And that's because the pentagonal faces were congruent.

The pentagonal base of the pyramid and the pentagonal cross section of the prism were the same.

And we were also told that the height of the prism was the same as the height of the pyramid.

So it's very similar to that triangular prism and the triangular pyramid, that there's always this relationship of a third.

So the formula for the volume of any pyramid, and hopefully you've seen this being established from going from a square-based pyramid to a rectangular-based pyramid to a triangular-based pyramid, is always calculated by doing 1/3 times the base area times the perpendicular height.

And you can think of that as a third times the volume of a prism, where the base area is equal to the cross section.

So this oblique hexagonal-based pyramid has a perpendicular height of 15 centimetres and a base area of 42 square centimetres.

So, what's its volume? Well, we need the base area, which we know, and we need the perpendicular height, which we know.

So we can substitute that in 1/3 times 42 times 15 is 210 cubic centimetres.

So, check for you.

What is the volume of this rectangular-based pyramid? Pause the video, and then when you're ready to check, press Play.

So, third times the area of the base, it's a rectangle, so you do that by doing length times width, multiplied by the perpendicular height, which is 15.

So, 490 cubic centimetres.

So let's have a look at some questions where we don't just get given the area of the base for example.

This square-based right pyramid has a base area of 50 square centimetres and a lateral length of 13 centimetres.

So you can see the lateral length marked on the diagram.

Sofia says the volume of this pyramid is a third times 50 times 13.

216.

66 Or 216 and 2/3 cubic centimetres.

Is Sofia correct? So, no, the lateral length of this pyramid is not equal to its perpendicular height.

The volume of a pyramid is the third times the area of the base, which is 50, times by the perpendicular height.

And in this case, the perpendicular height is not equal to the lateral length.

So how are we going to work out the volume? Well, the perpendicular height is an edge of a right-angle triangle with a hypotenuse of 13 centimetres.

So I've just marked it there on the diagram.

The other edge of the right-angle triangle is half of the diagonal length of the square face.

So we were told that this is a right pyramid, which means the apex is directly above the centre.

So that centre point is where the diagonals intersect on a square.

The square has an area of 50 square centimetres, we were told that, so the edges of the square are five root two.

We can square root it to get the edge length.

So now, if I look at the square face only, I would like to work out the diagonal length in order to be able to half it.

And I can use Pythagoras' theorem because a right angle is there in a square.

So Pythagoras' theorem can be used.

We've got five root two being the edge length of the square.

We got that from the area of the square.

So using Pythagoras' theorem, we can work out that the diagonal length is 10.

So for the purpose of our right-angle triangle, we're gonna half it and that will be five.

So we've now got a right-angle triangle that includes the perpendicular height, the lateral length that was given, and we've calculated half the diagonal.

So Pythagoras' theorem can be used substituting it in, we want to work out the shorter side.

So rearranging it, we can get the h squared is 144, and therefore h is 12.

The volume can now be calculated because we've got the area of the base, 50, and we've also got the perpendicular height, 12.

So 1/3 times 50 times 12 is 200 cubic centimetres.

So here's a check.

Here is a square-based right pyramid.

What is the perpendicular height of the pyramid? Pause the video, and then when you're ready to move on with the check, press Play.

So you're gonna use Pythagoras' theorem to work out that the perpendicular height, which I've called h, is eight.

So here is the same pyramid.

What is the area of the base? Pause the video and when you're ready to check your answer, press Play.

So the area of the square is 72 square centimetres.

You may not have done it in the same way, but I've looked at this from a plan view of the base, so it's a square, and I know from the diagonals intersecting and the diagonals dissecting each other that I get four congruent right-angled isosceles triangles.

And so I can do 6 times 6 divided by 2 to get the area of one of them and times it by four.

So that's 72.

So what's the volume of the pyramid? Pause the video, and when you're ready to check, press Play.

Well, the volume is a third times the area of the base, which you just calculated as 72, times by the perpendicular height, which you worked out as eight.

192 Cubic centimetres.

So onto the first task of this lesson, questions one and two are on the screen here.

So pause the video, read through them carefully, And then, when you're ready for the next questions, press Play.

Questions three and four are both on the screen.

Question three has got three parts to it.

Question four, there is just the one.

So pause the video, work through those, and when you press Play, you've got one more question of this task.

Here's question five, which is to work out the volume of this square-based right pyramid.

Pause video.

When you press Play, we're gonna go through our answers to Task A.

So question one, very wordy, but really, it was about this idea of the connection between a prism and a pyramid and their volumes.

So a hexagonal prism with a regular cross-section has a length of 23 centimetres.

The prism is filled with liquid.

How many hexagonal-based pyramids can be filled if the hexagonal base is congruent to the hexagonal cross-section of the prism, and the perpendicular height of the pyramid is 23 centimetres? Well, three pyramids would be able to be filled because one pyramid is a third of the prism.

Question two, an oblique pyramid has a base area of 36 square centimetres and a perpendicular height of 15.

What is its volume? Well, we've got the information we need, so it's just about substituting it in, 1/3 times the base area times the perpendicular height is 180 cubic centimetres.

Question three, as I said, there were three parts.

So you were given the base area and the perpendicular height, you just needed to work out their volumes.

So a third times 21 times 4 is 28 cubic centimetres.

For part B, it's 71.

7 to one decimal place.

And for part C, it's seven cubic centimetres.

Question four, there was no diagram.

Work out the volume of a rectangular-based pyramid with a length of 12, a width of seven, and a perpendicular height of 11.

So you needed to work out the area of the rectangle.

So 12 times 7, times it by the perpendicular height and times it by a third.

The volume was 308 cubic centimetres.

Lastly, question five, I've gone onto two slides to get through the solution.

So work out the volume of this square-based right pyramid.

So the perpendicular height, which I'm calling h, can be calculated using Pythagoras' theorem, and the right-angle triangle that you can see.

So the perpendicular height of the pyramid is 24.

The area of the square base can be calculated using four congruent right-angled isosceles triangles.

So 4 times a half times 10 times 10 gives us an area of 200.

So the volume can be calculated knowing that the area of the base is 200 and the perpendicular height of the pyramid is 24.

So the volume is 1,600 cubic centimetres.

So we're now onto the second learning cycle where we'll be making use of the volume.

This square-based pyramid has a perpendicular height that is equal to the edge length of the square.

Given that the volume of the pyramid is 72 cubic centimetres, what is its perpendicular height? Well, we can set up an equation.

We can use the formula for the volume of a pyramid, which is a third times base area times perpendicular height, and make it equal to the numerical value, which is 72.

We can simplify that algebra to a third a cubed, and we know that equals 72.

And we can then solve this to find that the perpendicular height of this pyramid is six centimetres.

On this, I'm gonna go through one myself and then you'll do a very similar one as a check.

So the pyramid has a volume of 168 cubic centimetres and a rectangular base with a width of seven centimetres and a length of eight centimetres.

What is the perpendicular height, h, of the pyramid? So we know the volume, we know the dimensions of the rectangular base.

So we have a formula for the volume of a pyramid.

And so we're gonna set up an equation.

So 1/3 times the area of the base, which is a rectangle.

So length times width times by the perpendicular height, which is what we want to find out, is equal to the given volume.

I can simplify the algebra to 56 thirds h equals 168.

I could write that as 56h over three.

I've timesed both sides by three, the using inverse operations, and then solve it to find that the perpendicular height of this pyramid is nine centimetres.

So here is a similar question for you.

Pause the video.

When you're ready to move on, press Play.

So again, yours was a rectangular base, different dimensions, set up the equation and solve it.

And your perpendicular height was 12 centimetres.

Continuing in the same way of I'm gonna go through one and then you can try one.

A hexagonal-based pyramid has a perpendicular height of 15 centimetres and a volume of 310 cubic centimetres.

What is the area, which I'm gonna call a, of the hexagonal base? So we've been given the volume, we have a formula for the volume, we have the height, which is one of the variables within the formula.

So a third times the area times 15 equals 310, which simplifies to 5a equals 310.

So a would have to be 62.

The hexagon's area is 62 square centimetres.

So here's a similar one for you.

Pause the video, and when you're ready to check your answer, press Play.

So again, set up the equation with what you know and what you want to work out, and solve it.

And the pentagon's area is 71 square centimetres.

Here we've got Jun and Alex, and they're discussing the volume of various pyramids.

Jun says, "The volume of the left pyramid is greater than the right one." Alex says, "How do you know that? Did you work them out?" Do you agree with Jun? Do you think he worked them out? June said "No, the base area is the same.

So I just compared the height." Yes, so because the formula to work out the volume is a third times base area times height, the third is a constant for both of them.

The base area in this case is the same.

So we just need to compare the heights.

If you're multiplying by a larger value, then the volume will be bigger, it'll be larger.

So here's a check.

The perpendicular height of all of these rectangular-based pyramids are the same.

Which of the pyramids has the smallest volume? Pause the video, and when you're ready to check, press Play.

So A has the smallest volume.

We don't have the perpendicular height, but we know that the perpendicular height is the same in all three of these pyramids.

So it's a case of working out who has the smallest area, because the third and the height is the same, so it's going to be the area that affects the difference of the volume.

4 Times 3 is 12, compared to 7 times 2 which is 14, compared to 5 times 3 which is 15.

So we're onto the last task of the lesson.

There's questions one, two, and three on the screen here.

There are no diagrams within this task, but if you want to sketch yourself a diagram 'cause you find that helpful, then, by all means, do.

But some of the diagrams might be quite a challenge to sketch.

Pause the video and when you're ready for the next questions, press Play.

Again, questions four and five are on the screen.

Pause the video, read them carefully, have a go at them.

When you're finished with those two questions, press Play 'cause you've got one more in this task.

So question six, there are three parts.

In each question, you are looking for the greater volume and you need to justify your answer.

So pause the video and go through question six.

When you press Play, we're gonna go through our answers to Task B.

So question one, a pyramid has a volume of 96 cubic metres and a polygonal base with an area of 48 square metres.

What is the perpendicular height? The perpendicular height would be six.

So it doesn't matter what shape the base is because you were given the area and it was the perpendicular height of the pyramid that you were trying to calculate.

Question two, a pyramid has a volume of 84 cubic millimetres and a perpendicular height of 18 millimetres.

What is the area of the polygonal base? So once again, it doesn't matter what shape that is.

If you were sketching them then you could have sketched any shape you wanted as long as it was a polygon.

So 1/3 times a, which is our area, times 18 equals 84.

So a is equal to 14.

So our polygonal base has an area of 14 square millimetres.

Question three had two parts.

A rectangular-based pyramid has a volume of 320 cubic centimetres and a perpendicular height of 15 centimetres.

So part A, which is the area of the rectangular base? So set up the equation and solve it.

The rectangle's area would be 64 square centimetres.

Part B, given that the width of the rectangle is four centimetres, what is the length? So, if we now know that the area is 64, 4 times what gave you 64? Well, that would be 16.

So the length of the rectangle is 16 centimetres.

Question four also has two parts.

A triangular-based pyramid has a volume of 540 cubic centimetres and a perpendicular height of 18.

Part A, what is the area of the triangular base? So set up the equation to find the area of the triangular base, and that would be 90 square centimetres.

Part B, given that the perpendicular height or the altitude of the triangle is twice the perpendicular height of the pyramid, calculate the base length of the triangle.

So firstly, how do we calculate the area of a triangle? Well, it's half times base times perpendicular height.

So we've been told that the perpendicular height of the triangle is twice that of the pyramid.

The perpendicular height of the pyramid is 18, so 2 times 18 is 36.

So if we set up our equation, a half times 36 times b, where b is the base, equals 90, and then we can solve that to see that b equals five.

So the base of the triangle is five centimetres.

Question five, a square-based pyramid has a volume of 1,331 cubic centimetres.

The perpendicular height of the pyramid and the edge of the square are in the ratio three to one.

What is the perpendicular height of the pyramid? So here, we need to think about our work with ratio.

So the perpendicular height and the edge of the square are the ratio three to one.

So I've let the edge of the square be a, so therefore, the perpendicular height would be 3a.

Now I can set up my equation.

1/3 Times the area of the square, which would be a squared, times the perpendicular height, which I know using the ratio will be 3a, equals 1,331.

Simplifying that algebraic expression gives us a cubed.

And so a cubed is 1,331, therefore a, by cube rooting it, a is 11.

The perpendicular height of the pyramid is 33.

'cause remember, a was the edge length of the square, and we have that ratio three to one.

Question six.

So there was three parts and it was always which has a greater volume, and in each case, justify.

So watch has a greater volume.

Pyramid 1, which has a base area of 14 square centimetres and a perpendicular height of 23 centimetres, or pyramid 2 because it has a base area of 13 and a perpendicular height of 23? So pyramid 1 would have the greater volume as the base area is greater than that of pyramid 2, whereas the perpendicular height is the same in both.

If you did this by just calculating the volumes to compare them, then there are written on the screen.

Part B, which one has a greater volume? Well, pyramid 2 would have the greater volume as the perpendicular height is greater than that of pyramid 1, whereas the base areas is the same in both.

So the base area is not going to affect one being bigger than the other because they are the same.

So we need to compare their perpendicular heights.

Once again, if you did just calculate the volume, they're on the screen.

And lastly, part C.

A cuboid or a pyramid.

Well, they actually have the same volume.

The perpendicular height of the pyramid is three times the length of the cuboid.

And as the cross-section and pyramid base have the same area, the volumes are the same.

So remembering that this idea of the prism and the pyramid, that the prism is three times the volume of the pyramid when they have the same height or length.

But here, the pyramid has three times the height of the prism.

So that sort of cancels out three times the third is the one.

Again, if you calculated the volume, they both come out as 240 cubic centimetres.

So, to summarise today's lesson, which was on the volume of a pyramid.

Pyramids are a group of 3D shapes.

They have a polygonal base and triangular faces that meet at an apex.

When that apex is directly above the centre of the base, it is a right pyramid.

The volume for any pyramid can be calculated using this formula.

So 1/3 times base area times perpendicular height.

And make sure that you know the perpendicular height.

If the volume is known, then the perpendicular height or the base area can be calculated, so we can set up equations and solve to find various things out.

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