Lesson video

Hello.

My name is Mrs Buckmire.

And today I'll be teaching you about volume of prisms and cylinders.

So first, make sure you've got a pen and paper.

Pause if you need to grab anything.

And I just want to remind you that you can pause the video whenever you like.

I'll tell you to pause at certain points, but also if you need to pause to give yourself more time, please do.

And if you need to rewind it because you want to hear that bit again, that's absolutely fine, okay? Go at your own pace.

Okay, so your Try this task.

Now, Binh has said, "It's not possible to find the volume of this cuboid, because you don't know the other two dimensions." And you'll see the cuboid there.

And Zaki says, "Mmm.

I'm not sure about that." Pause the video and have a think.

Who do you agree with? Do you think you can find the volume of this cuboid? Okay.

What did you think? You can find the volume? What is volume? Yeah, volume's a space, how much space the solid cuboid takes up.

So can you find the volume here? You guys know how to find the volume of a cuboid.

When you're given, let's say this one was eight centimetres.

Some of you guys might have been able to work out.

What am I going to say this one is? Good.

Nine centimetres.

So if this one was eight centimetres, this length, and this one was nine centimetres, then it's true.

The area of this front face is 72 centimetres cubed.

So then what would we do? We multiply it by? Yes.

A hundred, to find the volume.

So it'd be 7,200.

What number is that? Excellent.

7,200 centimetres cubed is the volume.

So Binh was right.

You could work it out.

So we didn't actually need these numbers at all.

I know it might not even be correct.

It might mean 72 and one, we actually don't know, okay? But I was just showing that as an example, but it was correct.

Okay, so this middle one here is the volume that we found out just now.

This one, you know how to find the volume 'cause you know that if the shape has been constructed with one centimetre cubed unit cubes, you can actually count the number.

And remember, you might have seen also that if we count kind of the front, so one, two, three, four, five, six, and then see how many rows.

One, two, three, four, five, six, seven, eight.

Then what would you do? Maybe you already know? Six times eight.

What's six times eight? 48 centimetres cubed.

That's the volume.

So we can find the area of that front and multiply it by the whole length.

And that would be a quicker way than just counting individually.

So what about when there aren't the squares? So here again, we've got these lengths and you know, with the cuboids we can do this area.

So eight times three, which gives you 24 and then times it by 10.

So it'd be 240 centimetres cubed.

So finally then, what about this one? You haven't seen this before? Have a think.

Good.

The volume, now, the area is actually given.

So just 36 times 10, so it'd be 360.

What the units? Centimetres cubed.

Good job! Have a think then.

What do you think all of these have in common? What type of solid shape are they? Yeah, they're all prisms. So do you know formula for the volume of a prism? What would it be? The volume of a prism equals the cross sectional area times the length.

What's the volume of this prism? Say it again.

Good.

It equals to 15 times six.

Six times five is 30.

Six times 10 is 60.

So it's 90 centimetres cubed.

Okay.

What about this one? I'll write that back up for you as well, actually.

You ever think about how that one's different? So how is it different? It's not a prism, but you can work it out in the same way.

So volume of cylinders equals the cross sectional area times the length, because that cross section area is always a circle.

So because it's consistent throughout the shape, we can actually work out the area of the circle and times by the length so really right from that cross-sectional area for a cylinder, you can't think of it as, "Oh, it's the area of the circle" because it always will be a circle.

But actually I personally, I always use cross-sectional area times length, even for cylinder.

But if you like, you can think of it like that.

Area of circle? What was it again? Good.

Area equals pi r squared.

What's r? The radius.

So here r is four.

So if we write this in terms of pi, the area, it would be pi times four squared.

Which equals to pi times four times four, which equals to 16 pi.

That's centimetres squared for area.

So what's the volume? What do you multiply the area by? Good.

Three.

So the volume equals three times 16 pi.

Three times 16.

So three times six is 18.

Three times 10, yes, 30.

So it equals 48 pi centimetres cubed.

Okay.

If you'd like to pause the video and write that down, so write either key formulas down or the examples, feel free to do that now.

Okay.

Quick reminder.

So I want you to match the shape to the area of formula and identify which lengths could be used to find the area.

So just pick them out.

You don't have to write anything down.

You can just think about this, if you need to.

Have a go.

Okay, so the first one.

Base times perpendicular height divided by two, which one's that? Good.

It is our triangle.

So this matches up to A.

So what is the base? I'm going to let this be the base.

So what's the perpendicular height? Good.

It's this one and let's call it perp height.

Hope you know, perp is perpendicular.

Okay.

So this one's not needed and this one's not need.

They're the two lengths that we need.

What about B? Base times perpendicular height.

Which one's that.

Good, it's this shape.

What's this shape called again? A parallelogram.

So a base and perpendicular height.

If I take this through my base we have perpendicular height.

Good.

Let's write it again.

It is this one.

Rather than if you got that.

So we don't actually need this one.

So often in life, there'll be things and added information that we don't need.

And as mathematicians, we need to decide, "Oh, what's important to what we're trying to do?" Okay, so that's just a skill that I want you guys to be practising here.

And then finally C, well not finally, one more.

Ah, C is not any of them.

Maybe you've used this as volume of cuboid.

And D? Yeah, I actually just showed you that.

So this one's D, pi r squared.

Which length is it? Good.

This is the radius here, r.

Now this length here, the name of it? Yeah, it was diameter.

How do you get from the diameter to the radius? Good, you could half it.

Well done! Okay.

You're ready now to your full independent task.

So the first bit here, just you applying those formulas.

So do go back and pause the video and write them down if you need to.

And then for 2 you'll find the volume of different prisms and in 3 I've given you a net.

I want you to work out the volume, okay? It might be helpful to do a little sketch for that one.

Do you pause the video and have a go.

I recommend actually going onto the worksheet so you can better see these questions.

Okay.

So feedback, the area of this shape, Now, we know the area of a triangle is base times perpendicular height.

So three is the base for this perpendicular height.

We don't need five, in this case.

Three times four, divided by two gives you six centimetres squared.

The next one is a, what shape? Yeah, a parallelogram.

So it's base times perpendicular height and that's it.

So it's going to be three times four.

And your final answer should have been 12 centimetre squared.

Okay, the circle.

It tells you the radius is two centimetres.

What's the formula again? Let's write it down.

A equals, yeah, pi r squared.

So the radius is two.

So here, our area equals to pi times two squared.

So that equals the four pi centimetre square.

Now, even though there's a pi, we still have centimetre squared.

Remember, pi is a number.

So it's 3.

1415, blah blah blah blah blah.

I'm not even sure if I got that completely right, but it is still a number.

So this represents a number and then the centimetre squared is our units part because it's then area.

So now I find the volume of each prism.

So what's the formula to find volume again? Good.

It's area of cross section times length.

So here, this is my cross section that's consistent throughout my prism.

And so we do four times three divided by three to find the area, times the length, which is six.

We did that above, how handy? So actually it's six times six, which equals 36 centimetres squared.

So remember, you must have divided it by two for this question as well.

Find the area of the triangle and divide it by two.

For this one, so the base here is three and the height is four.

So 12, again, is the area of the cross section.

And this is the length.

So if you cut like here, you would see a parallelogram.

That's the same as this one.

So that's why that's the length that it goes along.

So you do eight times four, sorry.

12 times eight, which is 96 centimetres cubed 'cause 12 is the area of the cross section.

And finally for the cylinder.

So cylinder is the area of the circle times the length, which is the area of the cross sections, this is the exact same thing.

And you get four pi times eight, so 32 lots of pi centimetres cubed.

And can you see all of these volumes, centimetres cubed.

Again, so for this one.

Now, I think it's super helpful to do a sketch.

So I would have sketched it like this, the 3D shape, and I've labelled this six centimetres.

So what will the height be? Good, it would be eight centimetres.

And what about the diagonal height, length here, sorry? Good.

That would be that 10 cm 'cause this part matching up to here.

So that's that 10 centimetres, and this length here? So that's this bit which matches onto this bit 'cause they join up to get you to two centimetres and then now we can find the volume.

So the volume is this area times the length.

So this area is going to be six times eight and then yeah, divide it by two, don't forget, which equals to 24 centimetres squared.

So the volume equals the 24.

And what length is it? Is it 10 or two? Good, it was two.

So 24 times two, which equals 48 centimetres cubed.

Well done if you've got that.

Okay.

So for your explore task, I want you to suggest lengths of each pair of shapes that would give the same volume.

So you're going to choose two of these 3D shapes here and you're going to say, "Oh, what the length would be, so they have the exact same volume?" Now, if you feel super confident, maybe you can find a length so all four have the same volume, okay? That is your challenge.

You can pause the video now, if you don't want to see Antoni's example, Otherwise just hold on.

Okay, Antoni says, "If shape 2 is five centimetres long and shape 4 is three centimetres long, they have the same volume." Right, let's just quickly verify that.

We don't want to just believe him.

So shape 2 is five centimetres long.

That's five centimetres and shape 4 is three centimetres long.

So what will the volume of shape 2 be? Well, 12 times five, which equals 60 centimetres cubed.

What would the volume of shape 4 be? 20 times three, which is 60 centimetres cubed.

True, they do have the same volume.

So, now you need to think of other numbers where maybe like to make 1 and 2 have the same volume.

To make 1 and 3 have the same volume.

To make 3 and 2.

Just see how many different ones you can come up with.

And if you want to have a go at the challenge, excellent.

Push yourself to have a go.

Okay, there are lots of different answers that you could have come up with.

So for 1, if 1 had a length of 12 centimetres, then the volume would be 96 centimetres cubed.

And then if 2 had a volume of eight centimetres, we know that 12 times eight is the same as eight times 12? What about if 1 had, let's say, three centimetres in length? Sorry, that was an L for length, but I'll just write three centimetres.

And then if 2, so eight times three is 24.

So 2 could have had two centimetres.

So even between 1 and 2, there's lots of different lengths you could have.

Hmm.

Did you notice something about common multiples? Yeah, we're just trying to find different common multiples.

So here the lowest common multiple on eight and 12 is 24, which is why three centimetres and two centimetres works out.

I wonder if you guys thought of that? Did anyone do the challenge? So you want all four to have the same volume.

So there are different answers.

I'm going to try and find the lowest answer.

The lowest answer, where they have the same volume.

If you didn't do that, pause and you can maybe have a quick go.

Okay, I think it is 120 centimetres cubed.

So for it to be 120 centimetres cubed for our challenge, then that means that, so 120, so this one's easy enough.

This one would have to be 10 centimetres.

The 21, 20 times six equals 120 centimetres cubed.

15, well, 15 times two is 30 and 30 goes into 120 four times.

So it would be eight centimetre.

You can try it out, test it, make sure I'm right.

Do eight times 15.

Pause it and check.

Is 120 a right or a wrong? And finally, eight times what equals 120? Eight times 15.

Ah, we just did it here.

Heya, these two are related.

15 centimetres.

So using these lengths, then actually, all of them would have the exact same volume.

Maybe from here, you could find even more lengths that could give them all the same volume.

Really, really well done.

If you had a go at the Try this, you listened on the connect and made notes.

You had a go at the independent task, explore task.

Then you know what? I think you've done a fantastic job today, okay? Lots of different answers you could come up with at the last task as well.

So if you want, do go over it again and check it.

Do you make sure you have a go at the exit quiz.

It's a really good way of you checking your understanding and getting some feedback as well.

I look forward to seeing you in our next lesson.

Bye.