Lesson video

Hello there.

My name is Dr.

Robson and I'll be guiding you through this lesson.

Let's get started.

Welcome to today's lesson from the unit of 2D and 3D shape with surface area and volume including pyramid, spheres, and cones.

This lesson is called the surface area of a cone.

And by the end of today's lesson, we'll be able to calculate the surface area of a cone.

A previous keyword that will be useful during today's lesson is surface area.

The surface area is the total area of all the surfaces of a closed 3D shape.

The surfaces include all faces and any curved surfaces as well.

And here are three new keywords that will be introduced during today's lesson.

We'll unpack these definitions very early on the lesson, but feel free to pause the video if you want to read these definitions before we start and press play when you're ready to continue.

This lesson is broken into three parts and I'm going to start by looking at cones and their curved surface.

A circular cone is a 3D shape that has a circular base and a curved surface from the circular base that meets at a point called the apex.

And here's the apex on the cone on the diagram.

It's the pointy bit.

Throughout this unit, all cones that we will look at will be circular cones.

However, cones of other bases do exist.

There are three important lengths on a cone.

One of them is the radius of the circular base which is labelled R on the diagram.

Another is the height or the perpendicular height from the apex to the circular base and that is labelled H in the diagram.

And the third is the slanted length from the apex to the circular base.

But rather than going perpendicular to the circular base, it goes along the curved surface of the cone and that is labelled L on the diagram.

So let's check what we've learned by looking at this diagram with measurements given.

Use those measurements to complete each sentence.

Pause video while you do it and press play when you are ready for answers.

Let's look at some answers.

The slanted length is 37 centimetres long.

The radius is 12 centimetres long and the perpendicular height is 35 centimetres long.

Now there are two different types of cone.

One is called a right cone.

And with a right cone, a line perpendicular to the circular base that passes through the apex also passes through the centre of the circle.

The other type of cone is an oblique cone.

And with an oblique cone, a line perpendicular to the circular base that passes through the apex doesn't pass through the centre of the circle.

When it comes to right cones, the curved surface of a right cone is a circular sector that has been folded.

On this slide, there is a link to an interactive GeoGebra file that allows you to see a cone gradually unfolding and folding again and you'll see how it becomes a sector of a circle when it's unfolded.

And it looks something a bit like this.

For this cone, when we unfold it, the curved surface would look like this sector here.

Different cones are made out of sectors with different radii and different angles, and you can see that on the GeoGebra file.

As you adjust the height of the apex, it affects what the sector looks like.

For example, with this cone here, if you unfolded that one, the sector would look a bit more like this.

You can see that the angle inside the sector is greater than with the first example we saw on the left.

So let's look at this cone in a bit more detail now.

We've got the radius of the base labelled and we've got the length of the slanted height labelled as well.

The curved surface has been unfolded to create this sector here.

Let's consider how R and L relate to the sector.

Well, when the sector is folded, the two radii of the sector become the slanted length of the cone.

So that means their length must be equal to L on the cone.

Also, these two endpoints of the arc, they meet to make the circumference of the circular base.

So the length of the arc on the sector is equal to the circumference of the circular base of the cone.

So we could use that fact to express the length of the arc for the sector in terms of R.

That's because with the cone, the circumference of the circle will be 2 pi R.

And if the arc length of the sector would be equal to that, then the arc length would also be 2 pi R.

With these facts now, we can calculate the area of the curved surface of a right cone.

And we can do that by splitting the sector into multiple smaller sectors, rearranging them, and approximating the area of a parallelogram.

For example, if we split the sector into four smaller sectors, we could cut them out and rearrange them a bit like this.

Now the radius of the sector in the middle which is labelled L, don't forget that is the slanted length of the cone.

And we can see that with a diagram on the right, but it would be here.

It would also be here as well.

Now for the diagram in the middle, the arc length is 2 pi R.

So for the diagram on the right, this length will be half of it.

So, it would be pi R.

And you can see it looks a little bit like a parallelogram.

It's not quite a parallelogram.

It hasn't got straight sides, but it's an approximation of a parallelogram.

And we can make it more so by cutting that sector into even smaller sectors.

For example, if we cut it into eight sectors and rearrange them like this, we could label on our measurements again.

So that is L, and this would be half the arc length.

So it would be pi R.

And if we cut it the sector in the middle into even more smaller sectors, it would look a bit more like this.

And you can imagine how as we cut into even more and more sectors, it would look more and more like a parallelogram because the base will get straighter and straighter and we can see as we do that, the length which is labelled L will become even more so the height of the parallelogram.

Now the area of a parallelogram is base times vertical height.

So using these measurements here, we could write that as pi R, the base, times L which would be the height.

So let's start making our way back towards the cone.

If the area of the gramme is pi R times L and the sector in the middle has the same area, that area would also be pi R times L.

And that sector is the curved surface of the cone.

So the area of the curved surface must also be pi R times L.

Let's take a look at an example with some numbers now.

For a right cone with a slant height of 45 centimetres and a base of radius 30 centimetres, we can calculate the curved surface area in the following way.

We can think about how the slant height will be equal to the radius of the sector when the curved surface is unfolded And we can also think about how the circumference of the circle at the base of the cone will be equal to the arc length of the sector.

So the arc length would be 60 pi.

These measurements also fit onto our approximation of a parallelogram.

Its height will be 45 centimetres.

Its base will be half of the arc length which is 30 pi.

Therefore, the area of the parallelogram will be 30 pi times 45.

So would be the area of the sector because it's equal and so would be the area of the curved surface of a cone.

So it would be 30 pi times 45 which is 1,350 pi centimetres squared.

Let's check how well we understood that.

The curved surface of this cone is unfolded to create the circular sector you can see in the bottom left of the screen.

Map these statements to the algebraic expressions.

Pause video while you do that and press play when you're ready for answers.

Here are your answers.

It looks something a bit like this.

So which expression or expressions show the area of the curved surface of this cone here with these numbers? And you've got the formula for the area of the curved surface at the bottom left to help you.

Pause video while you choose and press play when you're ready for answers.

The answer is C.

Pi times 16 times 65, where 16 is the radius and 65 is the slanted length.

Here you have another cone.

The curved surface of this cone is unfolded into a circular sector which you can see on the screen, but the values aren't labelled with the numbers.

They are labelled with A, B, and C.

Find the values of A, B, and C in terms of pie wherever appropriate.

Pause video while you do that and press play when you are ready for answers.

The value of A would be 37 centimetres because the radius of the sector is equal to the slanted length on the cone.

The value of B will be 24 pi because that will be equal to the circumference of the circular base of the cone.

And the value of C would be pi times 12 times 37 which is pi times a radius of the cone times the slanted length of the cone.

And that will give you 444 pi centimetres squared.

Let's now do the same thing again but in the opposite direction.

The curved surface of this cone is unfolded into a circular sector, but this time you can see that the numbers are given on the sector and there are unknowns represented by algebra on the cone.

You need to find the values of the lengths and the areas labelled D, E, and F on that cone there.

Pause video while you do it and press play when you're ready for answers.

D will be 18 centimetres.

The slanted length on the cone is equal to the radius of the sector for when the curved surface is unfolded.

E which is the radius of the cone, that would be 10 centimetres.

You'll need to take the arc length and consider how that would be the circumference of the circle and then divide by pi and divide by 2 to get your radius.

And finally F, the area of the curved surface, that would be 180 pi centimetres squared.

It's equal to the area of the sector.

And here's one more question for you.

The curved surface of this cone is unfolded into a circular sector.

What is the area of the curved surface of this cone? Now, you have the formula for the area of the curved surface, pi RL.

But you'll notice you don't necessarily have, well, the radius on the cone.

You might need to use the diagram of the sector in order to figure out what the radius of the base would be on the cone and then you can find the area of the curved surface.

Pause video while you do that and press play when you're ready for answers.

Let's take a look at this together now.

We don't have the radius of the base of the cone, but what we do have is the arc length of the sector for when the curved surface is unfolded and the arc length is equal to 2 pi R.

So that means pi R must be equal to 20 pi.

Now if you want to, you could work out the radius by dividing by pi to get R equals 20 and substitute that and the 48 into the form you can see in the bottom left hand screen.

We don't necessarily need to do that because pi R is in the formula.

So you could substitute pi R into it.

So you've got 20 pi and then times it by 48 and that will give you 960 pi centimetres squared.

Okay, it's over to you now for Task A.

This task contains two questions and here is question one.

Pause video while you do this and press play when you're ready for question two.

And here is question two.

Pause video while you do this and press play when you're ready for some answers.

Okay, here are your answers to question one.

Pause video while you check these against your own and press play when you're ready for question two answers.

And here are the answers to question two.

Pause while you check these against your own and then press play when you're ready for the next part of the lesson.

Well done.

So far we've looked at how to find the area of the curved surface of the cone.

Now let's look at how to find the surface area of the entire cone.

A closed cone is made from two surfaces.

One of them is its circular base and the area of the circle is pi R squared.

The other is its curved surface and the area of that is pi times radius times the slanted length.

That's pi RL.

Now a closed cone is different to an open cone because an open cone doesn't have a circular base and is only made from the curved surface.

A bit like an ice cream cone.

When it comes to a closed cone, the surface area of a closed right cone is the sum of the circular base and its curved surface.

In other words, you find the area of the circle, you find the area of the curved surface, and you add them together to get the surface area of a closed right cone.

Let's take a look at this with an example.

In this cone, the radius is 16 centimetres.

it's perpendicular height is 40 centimetres and it's slanted length is 43 centimetres.

However, we will not actually need all three of those measurements when we are calculating the surface area.

Let's see why.

The areas of the surfaces of this closed cone are a circle whose area is pi R squared.

So in this case, it'll be pi times 16 squared which is 256 pi.

And also the curved surface whose area is pi times R times L, which in this case will be pi times 16 times 43 which is 688 pi.

And that means the total surface area of this closed cone would be 256 pi plus 688 pi, which gives 944 pi which can be written as 2,970 centimetres squared if we round round to three significant figures.

So going back to the measurements we used, we've used the 16, the radius of the circle.

We've used the 43 which is the slanted length of the cone.

But what we didn't need to use to find the surface area was the perpendicular height of the cone, the 40 centimetres.

So let's check what we've learned.

Here we have a closed cone with the measurements given.

And what you need to decide is out of these five areas, which of them are areas of surfaces that belong on this closed cone? And it may be more than one answer.

Pause video while you choose and press play when you're ready for an answer.

The answers are B and C.

B is the area of the curved surface and C is the area of the circular face.

Hence, what is the surface area of this cone? And give your answer rounded to one decimal place.

Pause video while you do that and press play when you're ready for an answer.

Your answer is the sum of those two areas which will give you 1,606 pi which gives 5045.

4 centimetres squared to one decimal place.

Here's a different cone.

Could you please work out the surface area of this cone and express your answer in terms of pi? Pause video while you do it and press play when you are ready for an answer.

Okay.

Well, you need to work out the area of the curved surface and the area of the circle and then add them together and that'll give you 11,400 pi centimetre squared.

Now so far I have been explicitly stating whether the cone is open or closed, but that might not always be the case.

Whether a cone is closed or open may depend on the context that the cone is used in.

For example, Sophia wants to paint the outside of a plain, white cone-shaped birthday hat for her sister's party.

That's very nice.

What is the total area of the hat that Sophia paints? Now we're going to answer this question shortly.

But before we do, think about how in this context would we have a closed cone or will we have an open cone and why.

Pause the video while you think about that and press play when you're ready to continue.

Well, a birthday hat won't have a closed circular face.

Otherwise, you wouldn't be able to get on your head and that would be a terrible hat.

Therefore, we only need to calculate the area of the curved surface for this cone in this context.

So, over to you then.

What is the total area of the hat that Sophia paints? And give your answer to three significant figures.

Pause video while you work it out and press play when you're ready for an answer.

Here's your answer.

You do pi times the radius of the cone which would be 8 centimetres, half of 16, and you times it by the slanted length which is 36 centimetres.

That would give 288 pi which is 905 centimetres squared.

Okay, it's over to you now for Task B.

This task contains three questions, and here is question one.

Find the surface area of each closed cone and give your answer in terms of pi.

Pause video while you do that and press play when you're ready for more questions.

And here are questions two and three.

Pause a video while you read through these and answer them and then press play when you're ready to look at some answers.

Okay, here are your answers to question one.

They are all closed cones.

So, you need to find the area of the curved surface and the area of the circle each time and then add them together.

And these are the answers you get.

With question two, we have an ice cream cone.

That is an open cone.

Therefore, we only need to find the area of the curved surface, and that would be 168.

55 centimetres squared.

Then with question three, we have a telescope case with a lid.

And therefore, it's a closed cone.

So, you need to find the area of the circle, the lid, and the area of the curved surface, and then add them together.

And that would give you 396 square inches after you've rounded it to three significant figures.

You're doing great.

Now let's move on to the third part of this lesson which is using the surface area of a cone.

So far throughout this lesson, we haven't really used the perpendicular height of a cone in any of the examples we've been given.

That's because we've been using the radius and the slanted length to calculate surface area.

But what happens when you don't know how long the slanted length is? Well, the slanted length of a right cone can be found if you are given the perpendicular height and the radius of the cone.

We can do that by using Pythagoras theorem.

That's because L, H, and R form a right angle triangle and Pythagoras' theorem can be used to find any one of these three missing limbs when the other two limbs are known.

And because L would be the hypotenuse in this right angle triangle, when we substitute into a formula, it looks a bit like this.

Let's look at an example.

Here we have a cone where we know the radius of the base is 9 centimetres and we know that the perpendicular height is 17 centimetres, but we can't calculate surface area yet 'cause we don't know what the slanted length is.

But the slanted length could be calculated by doing L squared equals H squared plus R squared and substitute in our numbers 9 and 17 to make it look a bit like this.

We can then work out 17 squared plus 9 squared to get L squared equals 370.

Square root both sides of our equation to get L equals 19.

2 centimetres when rounded to three significant figures.

Now we know the slanted length of the cone and we already knew the radius of the cone, we can calculate the surface area.

I'll do it with this calculation here.

The 9 squared pi is the area of the circle.

The 9 times root 370 pi is the area of the curved surface.

And you'll notice that we've used root 370 in our calculation here rather than 19.

2 because the 19.

2 was a rounded number.

If we use 19.

2, it would give an answer that is approximately right, but it wouldn't be accurate.

It is much more accurate to go back and use the root 370 instead.

Or if root 370 was the last calculation you did in your calculator, you could use the answer button instead when you type this into your calculator.

However you do it, you should get 798 centimetres squared after you round to three significant figures.

So here's one for you to try.

We've got a cone.

We are given the perpendicular height and the radius but what you need to do is work out the length of the side, which is labelled W, and give your answer to three significant figures.

Pause while you do it and press play when you're ready for an answer.

Well, you'll use Pythagoras's theorem, a bit like this, and then you'll get an answer of 81.

5 centimetres.

So based on that, could you please calculate the surface area of this closed cone.

That means it includes a circle.

And give your answer accurate as two significant figures.

Here's your answer.

You do pi times 32 squared plus pi times 32 times 81.

5.

Or you can go one better than that and rather than type it in 81.

5, you could type in the whole number before it was rounded.

You should get a surface area of 11,000 centimetres squared.

The slanted length of a right cone can also be found when we know the radius and the surface area of the cone.

The surface area of a closed cone has a form and formula.

It has A, the area, equals pi R squared, which is the area of the circular base, plus pi RL which is the area of the curved surface.

So let's take a look at that with an example like this.

We've got a cone where we have the radius of the base is 20 centimetres.

We have the surface area of the closed cone as being 4,397 centimetres squared.

And let's now use that information to calculate the slanted length, L.

We can do it by thinking about our formula for the surface area of a cone and substitute in the numbers we know.

We know the area.

So, the value of A.

And we know the radius.

That's the value of R.

If we substitute in our numbers, we get this equation and we can simplify it by performing the calculations we can to get this.

And then we need to rearrange this equation so we get L as the subject.

We can do that by subtracting 400 pi from both sides of the equation.

Now at this point, you could type in 4,397 subtract 400 pi into your calculator and get a single number there.

But just bear in mind that if you round that number, your answer will become less and less accurate as you perform more calculations.

So what might be more accurate is to not type anything in until the equation's been fully rearranged.

So the next thing we're going to do to get L by itself so it becomes a subject will be to divide both sides of the equation by 20 pi.

And that means L would be equal to this calculation here.

When we type it into our calculator, we can get 49.

98 centimetres rounded to two decimal places.

So, let's check what we've learned.

Here you've got a closed cone and its surface area is 66 centimetres squared.

Its diameter is 6 centimetres.

Now you've got the formula for how to find the surface area of a closed cone on the screen.

Please use all this information to fill in the blanks labelled A to D.

Pause video while you do that and press play when you are ready for an answer.

Okay, let's take a look at some answers.

You would have 66, because that is the area, equals pi times three squared, because three is the radius of the base, plus pi times three because that's the radius, times W because that is the slanted length.

Could you solve this equation to find the length of the slanted length? And give your answer to two significant figures.

Pause video while you do it and press play when you're ready for an answer.

Your answer is 4.

0 centimetres.

Okay, it's over to you now for task C.

This task contains four questions, and here are questions one and two.

Pause video while work through these and press play when you are ready for more questions.

And here are questions three and four.

Pause while you work through them and press play when you're ready for answers.

Okay, here are your answers to questions one and two.

Pause video while you check these against your own answers and press play when you're ready for more.

And here are your answers to questions three and four.

You'll notice with question four that you do get two answers after you've solved that quadratic, but you can't use the negative answer because you can't have a negative length which is why you've chosen four, and then that gives 8 centimetres for your value of W.

Pause while you check all these against your own and then press play when you're ready to conclude the lesson.

Fantastic work today.

Now let's summarise what we've learned.

A cone is a 3D shape that has a circular base and a curved surface that meets at a single point, the apex.

An important length on a cone include the radius of its circular base which we label R, its perpendicular height which we tend to label H, and its slanted length from the apex along its curved surface which we tend to label as L.

The curved surface of a right cone can be unfolded to create a circular sector.

The area of the curved surface of a right cone is pi times a radius times a slanted length.

That's pi RL.

And the surface area of a closed right cone is pi R squared plus pi RL, where pi R squared is the area of its circular base and pi RL is the area of its curved surface.

Well done today.

I hope you have a great day.