# Lesson video

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Hello, my name is Mrs Buckmire and today's lesson is about the surface area of cylinders.

Now, I'm really excited to be teaching you this.

I hope we're going to have loads of fun in the lesson.

So make sure you're ready with a pen and paper.

Pause if you need to go get that.

Please remember you can pause the video whenever you like.

So if you need to have some time to write something down or you want to copy something off the screen, please do pause it and do remember to rewind it to listen to anything again, it can really help to aid you understand.

So please do that as well.

Okay, let's go.

So for the try this, what can you say about this cylinder that fits exactly inside this cuboid? Now I've added a cross-section to help.

So a cross-section is kind of as if we sliced it and what would it look like? And it's kind of from, let me do a little arrow, like this direction.

If we look at this direction, that's what the cross-section is.

So that's just to help you.

And I want you to say anything any facts you can tell me about the cylinder.

If you have no idea what I mean, what do you want me to do, just a minute and I'll give you some further help.

Okay, less than a minute, here you go with some support.

You can tell me what is the length of the cylinder? What is the diameter or radius of the cylinder? And does it have more or less volume than the cuboid? Okay, pause the video now and answer those questions and any other extra pieces of information you think you can give.

Okay, so I'm going to answer those questions first.

Right, so what is the length? Well the length is the same as this length here of the cuboid.

So it is 12 centimetres.

What is the diameter or radius? So the diameter, what's the definition again? Good, it goes from one side of the circumference to the other through the centre.

So the diameter, this is the centre.

The diameter is from here to here and it is four centimetres.

So therefore, what's the radius? Well, remember the radius is from the centre to the circumference.

So the radius is what to the diameter.

Good, it's 1/2.

So the radius is two centimetres.

And does it have more or less volume than the cuboid? Well, the fabric fits inside the cuboid and you can see here, there are actually some space as well.

We could get some grains of sand or something in there then actually it is a less volume than the cuboid.

Now you could have told me more information.

Maybe you found the area of the cross-section.

Maybe you found the volume of the cuboid or the volume of the cylinder even.

Again, well done if you found any extra information.

Good job.

Okay, so here is our cylinder.

We do know how to find the volume, but today we're doing surface area.

And for that, we can actually link it to the net.

Now, do you know what the net of the cylinder looks like? What shape are the faces? Good, there's definitely two circles.

But what about that curved surface area? What would it be? Good, it's a rectangle.

Are you not convinced? Okay, if you have a piece of paper, take it and roll it up.

Can you see how it can create a tube? Yeah, let's have another look at this task here.

Okay, so let's have a closer look at this net using GeoGebra.

So thank you GeoGebra and thank you Tim, for making this awesome app.

So we can see this is the net.

I'm going to show you how actually it can be made into a cylinder.

So this is our cylinder and here you can roll it out to a rectangle and two circles.

You can also now worth noting xxx that if we change the height, it doesn't necessarily change anything about the circles.

So change the height here, or make it really, really short, it's still going to be a rectangle and two circles and the circles don't necessarily change size, but how do the circles relate to the rectangle? Let's make the height here and look really carefully.

So you kind of see kind of this black line going round.

What's the connection between that black line and the rectangle? Good, so it becomes the length of the rectangle.

So if we say the other dimension's the height, then this is the length.

And what is that black line going round the circle? It's the edge of the circle.

What's the name for that? Begins with C.

Yeah, it's circumference.

So the circumference of the circle is equal to that length of the rectangle.

So that is a really important dimension you see.

Look at that.

Feel free, you can give GeoGebra and cylinder nets even, a little Google if you want to play around with something like this.

But let's connect that all together now.

Okay, so let's put that together.

So on my cylinder, I now actually want to draw in the dimensions.

So he's my pen, here we go.

So I know this length is 12 centimetres.

Collect this bit here to here.

The radius, now the diameter was four centimetres.

So actually the radius is from the centre.

So the radius is two centimetres.

And I'll tell you why I'm using that in a moment.

So what other dimensions? So we now actually want to know this width of the rectangle.

And I said to you it's exactly the same as the circumference of the circle.

Oh, we need our formulas then don't we? So what is the formula for the circumference of a circle? You know this, come on.

Yeah, I'm hearing C equals pi d or C equals 2pi r, yeah? So C equals pi d.

So pi times diameter is 4pi.

So I'm going to leave it as 4pi.

I'm going to leave in terms of pi so 4pi centimetres.

That's why I see small there, oops.

Okay, so they're all the lengths.

Now I actually want to find surface area, okay? So I'm going to first find the area of one circular face.

Now one circular face, now here now it's looking at why I did the radius, the radius is needed for area.

What's area of circle? Again, another formula? No, it's not pi d, that's the circumference.

It's pi r squared.

Area equals pi r squared for the circles.

What's r squared? Yeah, radius times radius.

So pi r squared.

So it's pi times two times two.

So it's 4pi.

So the area of the circular face is 4pi centimetres squared.

So the area of the two circular faces 'cause I've got the one here and then I've got the second one here, so 4pi doubled is eight lots of pi.

Okay, then I need the area of the rectangle.

So this rectangle is going to be 12 times four, whoa, 12 times 4pi.

Don't forget your pi.

Okay, it's the circumference, which is pi times diameter so should be 4pi.

12 times 4pi is 48pi centimetres squared.

So we have 48pi centimetres squared.

Then the total surface area, add them all up, what do you get? Eight plus four pi, 4pi even plus 8pi is 12pi.

12pi plus 48pi is 60 lots of pi centimetres squared.

And that is the surface area of the cylinder.

So the key, key, key points, you need the area of the faces, and so the two circles use the formula pi r squared.

I remember that there were the two faces.

So in the 3D shape object you can only see the one circle of face, but remember there's a second one that's equal to it.

And then the rectangle we need to use with the height, which were given or the length, but then the other dimension of the rectangle comes from the circumference of the circle.

So circumference of the circle is pi times diameter, say it again, circumference equals yes, pi times diameter.

And that's what you need to use then to find that other length of the rectangle.

Again, don't worry.

We're going to practise this loads.

Okay, so our first quick check, which calculation gives a surface area of the cylinder? So what I would recommend is actually having to, going through the motions of working it out and then comparing it to the calculations here, okay? You don't actually have to work out the final answer.

Just try and think about which calculation best matches to it.

Okay, pause the video and have a go.

Okay, so how did we do? So the radius is five centimetres.

I'm going to just draw it and it just helped me out here.

And this length is 12 centimetres.

So first I'm going to find the two circles.

So it's pi r squared.

So pi times five squared and we'd need two of them.

So I'm going to also times it by two, and then there are two circular faces.

And then it's add on the rectangle.

Now, the rectangle is going to be pi times diameter to get the length of the rectangle and the height is 12.

So pi times diameter, diameter is 10 centimetres.

So it's going to be 10 pi.

So the area of the rectangle is going to be 12 times 10pi.

So plus 12 times 120 pi or 12 times 10 pi, okay? So which calculation does it look like? So two times pi times five squared seems correct, but the 12 times five, that is not the area of the rectangle.

So this one's not correct.

This one pi times five squared.

Well, that's just one circle.

No, that's not correct.

We've only got one circle here.

That's not right here, two times pi squared.

Yes, this one, two times pi times five times 12.

Well, the 12 could be the length that's why.

And we have pi involved that's good.

Now two times five is 10.

Oh, they've done two pi r plus circumference.

So two lots on the radius makes the diameter of 10.

So I kind of did that in my head, but yes, that is correct.

So this actually gets the same answer as me.

This is the same so this is correct.

C, C, C is right.

This one pi five squared times 12, that looks familiar.

That's the volume.

It asks the surface area, d is wrong.

It's surface area not volume.

Well done if you've got C as well.

Okay, so I said, we'll have some practise here it is.

So I want you to draw an accurately labelled net of a cylinder radius six centimetres and length 10 centimetres.

By accurately labelled what I do mean? Is kind of do a sketch of it.

Not really with ruler, not really drawn to scale, but then actually label all the dimensions that you know best as possible.

And then b, in terms of pi, find the surface area the cylinder.

So like I did, don't actually work out what pi is just do it as lots of pi, okay? And then two, this is a bit of a challenge, but I thought, you know, you've got this far, you guys can do this.

Find the height of this cylinder, which has a surface area of 200pi metres squared.

I've given you some information on the diagram, okay? Right, have a good go at this.

Okay then, so let's go for it.

So first I'm just going to give you a generic kind of net there and I'm going to label what I expect for an accurately drawn.

So the radius is six.

So I think here I'd write six centimetres and the length is 10 centimetres.

The length between my two circular faces is 10 centimetres.

And you also might've liked to put the length of this on.

So this is going to be pi times diameter.

So it's just equal to the circumference of the circle.

The radius is six of the diameter is 12.

Good, so this is going to be 12 pi centimetres.

In terms of pi, find the surface area of the cylinder, okay.

So the area of the one circle, what is that? Good, pi r squared so pi times six squared, which equals 36pi.

So two circles, we should, the two circle faces equals 36pi times two, which is 72pi.

And now the rectangle.

Rectangle equals, now I already found that length is super useful.

So 10 times 12pi which equals 120pi centimetres squared.

Add these two together so just the two circles, the area of the rectangle and we you get how many lots of pi? 192 lots of pi centimetres squared because it is area.

Well done if you've got that as well.

Okay, a bit of a challenge, find the height of this cylinder which has a surface of 200pi metre squared.

Hmm, okay.

So what do we know? We know the radius.

So we know the radius then we know the area of the circle.

So the area of the circle here is pi r squared.

Good, so it's going to be 25 pi metres squared.

With me the bottom one is also 25pi metres squared.

So that means a rectangle.

What will the area of the rectangle be if I know the circle is 25pi, 25pi? Yeah, the total is 200pi.

Good, so I can do 200pi, take away 50pi equals 150pi metres squared.

That is the area of my rectangle.

Now I'm trying to work out the height.

I don't know, I know this whole thing equals 150pi metre squared.

Do I know this length? Yes, is the circumference of the circle.

We know the radius so we can work out the diameter.

So the diameter is five times two is 10.

The circumference equals pi times diameter.

And just write this down to make sure, so pi times diameter so the diameter was two times five, which is 10.

So circumference is 10pi.

So that means that length is 10pi.

Ah, it's now 10pi times the height equals 150pi.

The height must be 15 metres.

Well done if you got that.

Now, if you're not sure, feel free to rewind that.

Listen to that again.

That was a tricky question but I believe you can understand this.

Well done if you've got it first time, impressive.

Okay, for explore task, we are going to look at each shape alone, which you can't see yet give me a second.

Fill the boxes with the numbers three, four, and five to make the maximum and minimum surface areas.

And then I want you to tell me, what do you notice? So here we have a cylinder.

It has a radius of blank centimetres and a length or height of blank centimetres, okay.

And when I say length or height, I'm referring to this dimension here.

Okay, and then for cuboid, the three dimensions are blank, blank, blank in centimetres.

Okay, so I just want you to put in these numbers at different places, work out the surface area and just tell me what's the maximum one you find what's the minimum when you find, okay? There are a few different combinations and there'll be some support on the next slide if you're not sure, but otherwise just have a go have some fun with it.

Okay, so just some support again, let me, I would like have multiple shapes and just make sure you label this being the length and then what's the radius? Yeah, label the radius as well.

So why we do it? Just put random numbers and so let's say this was four centimetres and this was three centimetres.

This was four, this was three, this was five.

So these are referred to this length, this length and this length.

Yeah, just put them in and then find the different surface areas, okay? There are a couple of different ones.

So let's say the four and three, that one might relate to this one.

So where we have four centimetre radius and three centimetre length and maybe next time you switched numbers.

So that becomes three and that becomes four, okay? That's what I want you to just have a go find in different cylinders surface areas and cuboids.

Okay.

So, let's look through it.

So I'm going to start with the cuboid.

So where it was three centimetres, four centimetres and five centimetres.

Now what did you find out? What actually didn't really matter how we arranged these lengths.

It could be 3, 4, 5, 3, 4, 3, 5.

It could be 3, 5, 4, okay? Because of how cuboid is, it requires three dimensions.

So that's, you're going to get the same answer, whether it's on that one or whether this one.

It's obviously they're not drawn to scale if this was three centimetres, this was five centimetres and this was four centimetres.

You're going to get the exact same answer.

So what answer was that? What was the maximum and minimum surface area of this cuboid? Good, I got that too, 94 centimetres squared.

Okay, so this is the interesting one now the cylinder.

Now I've made them big just so that you can see them better, okay? Now I found lots of different ones and actually I had a really good time trying this out.

So I've drawn in the radiuses just to help me or the radar, maybe I should say so I'm going to have one.

The smallest one I found had a radius of three centimetres and a length of four centimetres.

So what I did then is I so found area here.

So it's nine pi centimetres squared, found the conference.

So it's double three times two, so six pi.

And that means this ones is whole thing is 24pi centimetres squared.

And then you need to remember, 'cause I forgot the first time.

So you need to remember, it's not just nine plus 24, is nine pi times two plus 24pi.

So remember pi, remember that there's two circular areas.

So you have 18pi plus 24pi.

So the smallest one I got is 42pi centimetres squared.

And that is when we have a three centimetre and a four centimetre diameter.

So three for the radius four for the height.

Did you get smaller? No, I don't think there is smaller.

This is the smallest so check your work carefully.

What about the biggest, what's the maximum surface area you found? Okay, so what I got was this one as five centimetres and this one as four centimetres.

And when I did that, you'll never believe this.

Not unless you've done this.

90pi centimetres squared.

That's massive.

When you think that the cube was 94 centimetres squared, but this is 90 times pi, like pi, what's about pi again? What number is close to? Yeah, three.

So 90 times three.

So almost three times bigger than the cuboid.

Very, very big, very, very big numbers here.

And that's what I got as the maximum.

Well, I will love to see your work.

So, especially think with that cylinder cuboid tasks there's lots of different variation.

I'd love to see how you showed your working out.

If you would like to share your work with me, please ask your parent or care to share your work on Instagram, Facebook or Twitter, tag it @OakNational and hashtag LearnwithOak.

I really hope you've enjoyed today's lesson.

Remember to find the surface area of the cylinder, you need to find the area of the two circles and the area of the rectangle.

How do you find that missing length on the rectangle? So not the height, the other one.

Yes, you use the circumference of the circle, right, time to test your knowledge with the exit quiz.

The exit quiz is super useful because it gives you feedback.

So you can really see how well you understood today's work.

Have a good day.

Bye.