Lesson video

Hello, I'm Mrs. Lashley, and I'm really looking forward to working with you throughout this lesson.

So during today's lesson, we're gonna work out how to get the surface area for cylinders.

On the screen, there are some keywords that we'll use during the lesson.

They're are words that you should have met and should be familiar with from before, but I would suggest you pause the video, reread them, and just make sure that you're happy with all of them.

The word cylinder is a new keyword for us today and it's a 3D shape with a base that is a circle and a parallel opposite face that is identical.

A cross-section of a cylinder made parallel to the base will be congruent to the base throughout the solid.

So our lesson's got two learning cycles.

Firstly, we're gonna find the surface area of a cylinder to calculate the surface area.

And in the second learning cycle, we're gonna make use of the surface area to find out other unknowns.

So let's make a start on the first learning cycle.

So we've seen the definition of a cylinder and both Jacob and Izzy are describing what they think a cylinder is.

So they've said that a cylinder's got a circular face and that it is a 3D shape and both of their descriptions fit both of the 3D shapes on the screen, which is a cone and a cylinder, but we are only discussing the cylinder.

So look at those two different shapes.

What makes them different to each other? Because they both do have a circular face and they both are 3D, but why is one known as a cylinder and why is the other a cone? Well, the reason the cylinder is different to the cone is because it has a uniform cross-section throughout.

So its base is a circular face as is the cone.

But if you were to cut parallel to the base at anywhere along its length, the circle that you would see, which is known as a cross-section, would be exactly the same size as the base, whereas, on a cone, you would get another circle, but it wouldn't have the same size as the base, and that's the difference between the cylinder and the cone.

They both have a circular base.

They're both 3D shapes as both Jacob and Izzy said.

However, when we cut it and we slice it parallel to the base, the cylinder is where you'll get a uniform cross-section, whereas, on the cone, the cross-section is a different size.

During the lesson today, we're gonna be discussing cylinders a lot and we're always going to, during this lesson, be speaking about what's known as a right cylinder.

And most people, whenever they use the word cylinder, are also talking about right cylinders.

If it was an oblique cylinder, then they would tell you that.

So if we think about what a cylinder is, Alex has said "That's the same as a prism," and you can see there the definition of the prism.

Sophia has disagreed because she's saying a circle is not a polygon.

So that definition of a prism is that the base is a polygon, whereas, our cylinder, we have seen the base is a circle.

So although it seems very similar to a prism, the fact that the circle is not a polygon means that it is not a prism.

Over the next few slides, we're gonna look at a 3D solid, specifically prisms, and its net.

So here is a right equilateral triangular prism.

So we've got the view of the solid on the left and a version of the net on the right.

This is a square faced cuboid.

And again, you've got the net on the right-hand side.

So I'm gonna stop talking for a moment and we're just gonna increase the number of edges on the base face.

So we started with three, the equilateral triangle and then we've moved up to four.

It's now a square faced cuboid.

It's gonna be a regular polygon every time.

And I just want you to be thinking about what's changing, how that evolves on the net over the next few 3D shapes.

So this is a regular pentagonal prism, regular hexagonal prism, regular heptagonal prism, regular octagonal prism.

So at this point what has changed? Well, the base face is changing, but so on the net those faces are changing.

But rectangles are throughout on the net.

There's one more each time as the number of edges of the face is increasing.

This is a regular nonagonal prism.

Have a try saying that one.

So a nonagon is a nine-sided shape, and this is a regular nine-sided shape.

The base is a regular nonagon and it's a prism.

Hopefully, you can recognise that the length of all of those rectangles is actually the perimeter of the polygon that this prism is based around, and that's because those rectangles are wrapping around the perimeter of the polygon.

So here's a check on that idea.

If you know that the edge length is five, what would the length of X be? Pause the video whilst you calculate that and then press play and we'll go through the answer.

X would be 45.

So it's a nonagon.

That polygon that is the base of this prism is a regular nonagon.

So regular means all the edges are equal in length.

So you know that one edge is five.

A nonagon has nine edges.

So nine multiplied by five is 45.

So we're gonna continue working through these prisms, these regular prisms as the edges increase.

So here we're up to a 20-sided prism.

You can see the net.

You've still got those rectangles.

Those rectangles are still going to have a length, a combined length of the perimeter of the 20-sided polygon because they wrap around the outside.

We haven't continued from 20 up to 35.

We've just skipped to 35 now.

So this is a regular 35-sided prism.

Again, if you look at the net, we've got those two congruent faces that is the base and then its parallel, identical face, and then those rectangles that sort of making the colour in the way that it's sort of stood as a solid.

So hopefully you've been considering what's going on and maybe having some thoughts of your own, but so we've got some Oak pupils here that have also been thinking about what we've just seen.

So Laura has said, "As the numbers of edges are increasing," so that last one we saw was a regular 35-sided prism, "will it ever become a cylinder?" Izzy said, "It has started to look more like a circular face." When we've had the net, you can see the polygon, which is the base of the prism, and they have become a little bit more circular is what Izzy is saying.

But does that actually mean it will ever become a circle? So they're becoming more circular, but are they going to become a circle? Jacob believes that there's gonna be an angle that surely there will still be an angle between the edges in a many-sided polygon.

So polygon got straight edges, and there will always be an interior angle between two of the adjacent edges.

And he's saying if you zoomed in far enough surely there is an angle between them.

So let's take a look at that.

Here is a 50-sided polygon.

Maybe you can just about make out that there are straight edges, but it definitely is becoming more circular.

If you looked at that very quickly, you might believe that that was a circle, but it's actually a 50-sided polygon.

And if we zoom in into just one vertex, one of the 50 vertices, if we just zoom into one, there actually is a very large obtuse angle between the two edges, 172.

8 degrees.

So Laura's saying, "Well, if it becomes more than 50," so that was 50-sided shape, but we can have more edges than 50, "there's still gonna be an angle between the edges." Izzy says, "But it does help us think about what a cylinder's net will look like" As these polygons become more circular, they feel more circular.

What would the net look like for that? Well, hopefully you saw as the increased numbers of sides that the net wasn't changing very much.

You still had your identical base polygon and then you had lots of rectangles that were making up the sort of collar of the prism.

Jacob's referring back to the definition.

The definition of a prism is it has a polygon as a base, whereas a cylinder has a circle face.

So a circle is not a polygon, therefore it isn't a prism, but we can see that they're quite similar in their structures.

So a quick check on that.

As the number of sides of the polygon increases, it starts to become a circle, more like a circle or larger.

Pause the video whilst you're considering that and then when you press play, we'll go through the answer.

It will never become a circle.

Hopefully, that conversation between Laura, Izzy, and Jacob allowed you to see that.

Doesn't matter how many sides the polygon ends up having.

There will always be an angle between these two edges, whereas a circle hasn't got angles between edges because there is one curved edge.

There is not any straight edges to create an angle.

So it would become more like a circle.

The larger is dependent on the edge length.

If the edges were all one millimetre, it's not very big.

If the edges were all 20 metres, it would be a very large polygon.

So the larger part is nothing to do with the amount of sides.

It depends on the length of each side.

So Izzy said that we are gonna get an idea of what the cylinders net would look like.

So if you could unfold a cylinder, it would look like this.

We'd have these two circular face ends and they are.

That's the cross-section that runs throughout, and they're uniform in size, otherwise we'd end up with a cone.

So they're uniform in size, so we've got identical circles at either side.

And then the curved face which wraps around the circle would lay out and look like this.

So the curved face has become rectangular when laid flat and you could grab yourself any sheet of paper and you would manage to curve it round.

So assuming your piece of paper is rectangular, if you just sort of curved it with your hands, you can make a tube-like structure similar to a cylinder.

It wouldn't have its ends.

You wouldn't have the circular ends on there, but you would see that the curved surface when laid flat is rectangular.

And that length of the rectangle, and this is the important part here, the length of the rectangle is the perimeter of the circle in the same way that the length of the rectangles combined for the prisms was the perimeter of the polygon.

We just have a specific word for the perimeter of a circle, and that is the circumference.

So the length of the curved surface in its net is the circumference of the circle.

And then finally, the width of the rectangle is all dependent on the cylinder.

And it might be a length, it might be a height, and it could be known as a depth, and that's all to do with how the orientation of the cylinder.

So if you've sat a cylinder on its base on a circular face, then you're probably gonna talk about the cylinder having a height.

But if it was sort of laying on its curved surface, then you might talk about it having a length or a depth.

So just being mindful that you're gonna see those words, and then will mean the same thing.

We know what that net has been created from.

We know now where it's come from if you imagine unfolding a cylinder.

So now we're thinking about the surface area of the cylinder.

And just a quick reminder, what the surface area is.

The surface area is the total of all faces areas.

So in this case we've got two circular faces and the curved surface is now a rectangle in its net form.

So we need two lots of the circle area because there are two identical circles and then we need to add on, 'cause it's the sum of all of the faces areas, the rectangle area.

We've previously learned that circles.

Pi r squared is the formula for working out the area of a circle, and there are two of them.

So two times pi r squared.

And the rectangle, if you remember, the length of that rectangle is the circumference, and then the width of the rectangle is the height or the depth or the length, depending on the context, but the height of the cylinder.

But we have a formula for a circumference of a circle and that's two pi r.

We are now building the formula of the surface area to remove the words, the sort of where the parts are coming from and fit really writing it as a formula with variables.

And so this is sort of the finished product.

For any cylinder with a radius length of r and a height of h, the surface area can be calculated by doing two pi r squared plus two pi r h.

It's the total of the parts.

Two pi r is the circumference.

So that gives us that length of the rectangle, and pi r square is the area of the circle.

The reason we've got the coefficient of two is because we've got two identical circles.

As I say, that is the formula for any cylinder regardless of its height or its radius.

A quick check here.

Surface area of any cylinder is two pi r square plus two pi r h.

We've just seen that and hopefully you recognise the parts of it that we've got the area of the circle, the circumference of the circle, where are r is the radius and h is the height.

R and h are constants, variables, or terms? Pause the video whilst you consider that, and then press plate, and we'll go through the answer.

They're both variables.

They are things that will change depending on each cylinder.

The two pi are both constants, and then two pi r squared and two pi r h are both terms. Okay, so let's use the formula to find the surface area of this cylinder.

We've got the diagram there of the cylinder and so in this case we'd probably speak of the four being a height because of the way that the cylinder is orientated.

A quick sketch of the net of a cylinder so that we can really focus on what each part of the surface area formula is doing as we start to use it for the first time.

So taking the information off of the solid diagram, we've got our height of four and that will be our width of the rectangle.

We've got a radius of three, and the length of the rectangle, which is our curved surface, but when laid flat is rectangular, is the circumference, which is two times pi times three because three is our radius.

So now we can just substitute those values into our formula.

Two times pi times three squared is working out the area of the circle pi r squared and timesing it by two because there are two identical ones, and then we're adding on the circumference multiplied by four.

So pi times six is the circumference because two times pi times three is six pi or pi times six, and then multiplying by the heights.

Individually, those terms, 18 pi and 24 pi, it's worth keeping it in terms of pi.

So just evaluating the numerical parts and multiplying it by pi, so leaving it in terms of pi, that is the most accurate.

That's the most accurate way of getting to our answer so we don't include any rounding because pi is the irrational number that at some point we'd have to round if we wrote it in any other way except from using the symbol.

Because they are both in terms of pi, we can add together 18 lots of pi and 24 lots of pi, and so the surface area is 42 pi square centimetres.

As I say, in terms of pi that is the most accurate, unless the question tells you to give it to one decimal place or so many significant figures, I would always encourage you to leave it in terms of pi.

So we've worked through an example using the formula, but Lucas and Aisha have noticed that actually the formula that we use two pi r squared plus two pi r h has got equivalent formulae.

So Lucas has noticed that both of the terms have got a common factor of two pi R.

So he has factorised out two pi r and rewritten it as two pi r multiplied by r plus h.

And Aisha has noticed that it's equivalent to two pi r squared plus pi d h, and that's because circumference of a circle is also pi times diameter.

Diameter is two times radius and that's why two pi r and pi diameter are equal.

So here is another cylinder.

If Lucas works through the surface area for this cylinder using his equivalent formula and Aisha works through using hers, hopefully, you can see that they are equivalent.

Using Lucas's equivalent formula where he had factorised out two pi r, we've got this cylinder and radius of three millimetres and a height of five millimetres.

So substituting those in, it comes out as 48 pi square millimetres.

Aisha's going to use her version, and that was used in circumference pi d or pi times diameter.

So that one she needs to change the radius that was given, which was three, into a diameter by doubling it.

Again, you get 48 pi A check.

You have to use a certain formula, depending on if you have the diameter or radius of the circle.

True or false, and then justify your answer.

Pause the video whilst you're reading through, making your decisions.

And when you are ready, we'll go through what should have been the correct answer.

So it's false.

Although one might feel more useful in certain circumstances, you don't have to use it, and that's because there is a multiplicative relationship between the radius and diameter.

You are gonna do a bit of practise to work out the surface area.

So on this question, there are four cylinders with radius or diameter potentially labelled and you need to work out the surface area for each.

On D, be careful 'cause there is mixed units so you're gonna have to convert one of them.

Pause the video whilst you work through those.

And then when you come back, we'll go on to question two.

Here's question two.

So question two is more contextual.

So a drinks company are looking to use less material when making their cans.

These two cans can both hold 330 millilitres of liquid.

Which can should the drinks company choose? Pause the video whilst you have a go at this question.

And then when you press play, we're gonna go through the answers to question one and question two.

So question one A and B are both on the screen now.

I've given the answer both in terms of pi and rounded to one decimal place, depending on if you were using your calculator or not.

It may leave it in terms of pi for you, but it may have also given to you in a decimal form.

So I didn't want you to be concerned that you were incorrect or correct or not.

Here is part C.

So all in millimetres here.

Again, it's in terms of pi and also to one decimal place on the screen.

And then lastly on D.

So there were mixed units on D.

So I've given you the working out in metres if you converted the 90 centimetres to metres, but I've also given you the answer in centimetres in case you turned the metres into centimetres.

Question two.

It was a surface area question because they wanted to use less material to make the can.

The material is on the surface of the can.

How much it holds is to do with its volume and its capacity.

So this is a surface area question because it's about the material and not about how much liquid it could take.

So we need to do the surface area of both.

So here is the surface area of A.

Here's the surface area of can B.

And so the drinks company should go for the can B style because that uses less material.

The surface area is smaller, is less than the surface area of A.

So we're now gonna move on to the second learning cycle, which is using the surface area of a cylinder.

We're gonna use it to find other lengths, other unknowns.

In the first learning cycle, we developed the formula for the surface area of a cylinder where r is the radius and h is the height.

And so the formula involves three variables, the surface area, the radius, and the height.

We did also see that there are some variations of the formula, but ultimately they all mean the same thing because of the multiplicative relationship between the radius and diameter.

The surface area is dependent on the radius and height of the cylinder.

Andeep clarified that by saying, "Well, this means that if the radius was to change, the surface area would too." And that's why we say it's dependent on it.

So if the radius changes, then the surface area changes.

Similarly, if the height of the cylinder was to change, then the surface area would change as well.

The radius and height are not necessarily dependent on each other.

So in some questions you might be told that the height is twice the length of the radius, and then there is a connection between the two of them, but often there isn't a relationship between radius and height.

so they're not dependent on each other.

There's a link that takes us to a dynamic software.

You can click on that link and have a play with that dynamic software to highlight this idea, and I'm just gonna talk us through it now.

So here we've got a cylinder with the height currently set at five and a radius at 6.

56.

The surface area's been calculated to two decimal places as 476.

18.

I can move the sliders to change the height and I can move the slider to change the radius.

What we've just spoken through is that the surface area depends on the radius and also depends on the height.

So what we should see is that if I change the radius, the surface area changes.

If I change the height, the surface area changes.

But if I change the height, this doesn't change the radius because then they are not dependent on each other in this setup.

So I'm gonna change the radius.

And hopefully you saw there that the surface area was increasing when the radius got larger, and it was decreasing when the radius got smaller.

Now I'm gonna change the height.

Once again, as the height increased, the surface area got more.

And as the height decreased, the surface area got less, but the radius didn't change when I moved the height.

On this check, I would like you to fill the blanks in the sentences.

So pause the video whilst you figure out what they should be, and then press play, and we'll check to see how you got on.

Since the surface area is dependent on the radius and the length of the cylinder, it doesn't matter which way round you put those two words as long as you've got length and radius.

If the radius changes, the surface area will change too.

So because it is, the radius is an independent variable and the surface area is the dependent variable.

Here we've got a problem that's in context.

So a small cheese making company needs to wax a cheese truckle.

Truckle is just the word for the shape of a cheese when it is formed.

They have enough wax to cover an area of 1,120 pi square centimetres.

The cheese truckle has a radius of 14 centimetres.

We can see that on the cylindrical image there.

How high can the cheese truckle be? So they've only got a certain amount of wax to cover up the cheese truckle to keep it fresh, and we need to figure out how high they can make the cheese or where will they need to cut it.

So when we pose with a question like this, you need to think about what information do you actually have.

Maybe it's from a diagram or maybe it's from the words.

So what information do you have? Well, we know that the surface area has a maximum surface area of 1,120 pi because this isn't how much wax to cover the cheese.

So if we're covering the outside of the cheese, which in this case is cylindrical in shape, then it's a surface area.

We also know that the cheese has been formed into a circle of radius 14 centimetres, and therefore we can use our surface area formula, substitute the parts we have into it, and then solve it to work out the unknown.

And the unknown for this question is the height.

So where there would've been surface area, we've got 1,120 pi.

Then we've substituted the radius of 14 into the spaces where there would've been an r, 'cause remember it's two by r squared plus two pi r h.

The h is staying there.

That is the height.

That's the unknown that we have that we need to solve to calculate.

So we can calculate and simplify the first term to 392 pi.

We can simplify two times pi times 14, which we know is the circumference of that circle, and that's 28 pi.

We're gonna solve this equation by firstly subtracting 392 pi from both sides.

It's leaves us with 728 pi on the left-hand side and it has removed the term 392 pi on the right hand side.

Still we're working with in terms of pi.

We know that's the most accurate, and now we need to figure out what multiplied by 28 pi gives you 728 pi.

And by using inverse operations of division, we can work out that h could be 26.

In the context of this question, this means the cheese truckle can be 26 centimetres high.

I'm gonna go through an explanation, and then it's your turn to do a very similar one in a check.

So a cylinder has a surface area of 350 pi square centimetres and a diameter of 10 centimetres.

What is the height of the cylinder? So going through the thought process again, what do you have? Well, you have the surface area.

What else do you have? You have the diameter and you want the height.

The diameter and the radius are connected.

We know that the diameter is twice the length of the radius.

So for the formula two pi r h, we need to half the 10 to get our five that you can see there.

So two times pi times five squared plus two times pi times five times h equals 350 pi.

350 pi is that given surface area.

And we were given the diameter which implicitly tells us that the radius is five.

Simplifying anything we can and then starting to solve.

So we then get down to 10 pi h equals 300 pi, which means that h would be 30 centimetres.

If we divide both sides by 10 pi, then we are left with 30.

The height of the cylinder would be 30 centimetres.

So here's one for you to do.

Use my example to support you if you need it.

Pause the video whilst you're having a go, and then when you're ready to check your answer, just press play.

So you were given a surface air of 90 pi and a diameter of six.

So you needed to take that diameter of six and convert that to a radius of three.

Then you can substitute into the formula.

The unknown or the variable that you were needing to solve was h.

Simplifying it, hopefully, whichever order you went for solving, you should have got h to be 12 centimetres.

And notice that on your example, I called it a length of the cylinder, whereas on my example it's a height, but those two words can be interchanged depending on how the cylinder is sort of orientated.

So now you're gonna do some further practise of using the surface area of a cylinder.

So on this screen, you've got three cylinders.

So they're sort of three separate questions, but for each one, you're finding the missing length and make sure you give your answer in centimetres.

Pause the video whilst you have a try.

And then when you are ready for question two, press play and then we'll show you question two.

Question two.

A necklace is being made.

Part of the design is a metal cylindrical rod.

The rod is gold plated.

208 pi squared millimetres of gold leaf is used, the rod has a diameter of four millimetres.

How long is the rod? Pause the video whilst you work this one out.

Think about what information you have, use the formula.

And then when you're ready to check your answer, we'll go through question one and two.

Question one, this was the first one with the a length that you were trying to calculate.

So you were given the radius of three.

You were given the surface area of 78 pi, substituting that into the formula and rearranging it to solve it.

Hopefully you got a to be 10 centimetres.

The second cylinder with the missing height of b.

Again, this time the diameter was given, so you needed to transfer that into a radius of six before you substituted into the formula.

And b should have come out as eight centimetres.

The last one of this, the unknown length of the cylinder is c.

And you were given a diameter of 80 millimetres.

This one though needed you to do a couple of things.

You needed to recognise it was the diameter.

So first, you needed to half it to, get the radius, which would be 40 millimetres.

But you were also wanting to give your answer in centimetres.

The length was in centimetres.

The surface area was in square centimetres, so we needed to be consistent with the use of our units.

So 40 millimetres needed to then become four centimetres.

So it was a little bit of preparation before you could start calculating.

And c was seven centimetres.

Question two was a contextual question.

You needed to recognise it was a surface area question because they were gold platting this very small rod as part of their necklace design.

So if they were applying gold leaf around the rod, then that was again on the surface.

So you had the surface area.

The diameter was four, so you needed to change that to a radius of two, and then substitute it in.

And the length of the rod was 50 millimetres, so five centimetres in part of this necklace design.

So in summary, all cylinders have a circular cross-section so a formula can be derived for the surface area.

So unlike a prism where all prisms are slightly different, depending on their base, cylinders are all the same in terms of they have a circular cross-section.

So that's why we could derive the formula.

If the surface area is known, then other lengths can be found.

So we did that in the second part of the lesson so we can calculate the surface area, but we can also use it to work out missing lengths.

Well done today and I look forward to working with you again in the future.