Lesson video

Hello, everyone, I am Mr. Gratton.

Welcome and thank you for joining me in this lesson on the surface area of 3D shapes.

Today, we will look at different parts of a frustum and how we can calculate its surface area.

Pause here to have a quick look at the definitions of frustum and surface area.

First up, let's look at one important part of a frustum, it's curved surface.

Let's get straight into it.

Right, here we have a cone.

It has two surfaces, its circular base which has an area of pi r squared and its curved surface which has an area of pi rl where l is the slant length, the length along the surface of the cone from its apex to its base.

We can unfold the curved surface into a circular sector that has a radius of l and an arc length of two pi r.

For this part of a lesson, let's focus on just the curved part of the cone with an area of pi rl.

This cone has a curved surface of pi multiplied by seven multiplied by 13 which is 91 pi or 286 centimetres squared.

On to you, pause here yo find the area of just the curved surface of this cone.

Leave your answer in terms of pi.

Its radius is 11 centimetres and its slant length is 21 centimetres, so its curved surface has an area of 231 pi centimetres squared.

Here we have Sam who recalls that a frustum is a large cone with a small cone cut off and removed from it.

We know how to find the curved surface area of a cone, but how can we find the area of a curved surface of a frustum instead? Pause here to think about or discuss what we might need to do to find the area of a frustum's curved surface.

And Aisha correctly notices that the curved surface of the frustum is the curved surface of the large cone, take away the curved surface of the small cone.

Well done, Aisha, let's put your observations into practise.

However before we can calculate any areas, it is essential that we find the lengths of both radii and both the slant lengths.

For the large cone, the slant length is the sum of the slant lengths of the small cone and the frustum at 39 centimetres and its radius is 18 centimetres.

For the small cone, its slant length is 13 centimetres and its radius is six centimetres.

Now we know all of this information, let's look at the curved surfaces for both cones.

For the large cone, its curved surface is 702 pi whilst for the small cone, its curved surface is 78 pi.

The curved surface of this frustum is the difference between the curved surfaces of those two cones.

So we subtract these two areas to get 624 pi or 1,960 centimetres squared.

So 1,960 centimetres squared is the area of this curved surface of the frustum.

This does not include the areas of its two flat circular faces.

For this check, pause here to find the values of a to d to complete this table about that frustum.

A is 24, B is 20, C is 84, and D is the subtraction of 84 and 14 which is 70.

Using this information, pause again here to find the areas of these three curved surfaces.

Okay, the curved surface of the large cone is 2,016 pi, the curved surface of the small cone is 1,400 pi, and the curved surface of the frustum is 616 pi, the difference between those two other curved surfaces.

For this different frustum, Sam does not believe we can find the area of its curved surface since we do not know its slant length.

However Aisha believes that we can.

Pythagoras' theorem absolutely can be used to find its slant length, but what other information must we know? First, the radius and vertical height of a cone are important pieces of information to know.

This is because they are perpendicular to each other.

These two lengths can then be mapped onto the two shorter sides of a right angle triangle.

Because these two lengths can be mapped onto two of the three sides of a right angle triangle, Pythagoras' theorem can be used if given the vertical height and radius in order to find the slant length, the third missing length of that right angle triangle, and on the frustum and cones.

For the larger cone, we have this right angle triangle between the radius and vertical height of the cone.

Using Pythagoras' theorem for these three lengths with these values substituted in, we can then solve this equation to get L equals seven times the square root of 901 which is the slant length of the larger cone.

For the small cone, we have this smaller right angled triangle with a width of 60 centimetres, the radius of the smaller cone.

The height of the small cone can be found by subtracting the height of the frustum from the height of the large cone, giving 104 centimetres.

Using Pythagoras's theorem for these three lengths with these values substituted in, we can then solve for L to get L equals four times by the square root of 901 which is the slant length this time of the small cone.

Brilliant, now we have these two slant lengths, we can find the area of the curved surface of each cone like so, where the curved surface of the frustum is the difference between these two other areas of the two curved surfaces of the two cones.

The curved surface of this frustum has an area of 46,700 centimetre squared.

Okay, let's check your understanding by only reading values from this frustum diagram and maybe one bit of addition.

Pause here to find as many values of a to f as possible.

We have the radii of 80 and 20 and the vertical heights of 100 and 25.

Now we know the two radii and the two vertical heights, pause here once more to use Pythagoras' theorem to find the values of e and f.

Here are the two equations we can set up using Pythagoras's theorem for the two slant lengths of 128 millimetres and 32 millimetres.

And finally, now we know both slant lengths.

Pause here a final time to calculate the areas of the curved surfaces of the two cones and hence the area of the curved surface of the frustum.

The curved surfaces of the two cones are 32,170 and 2,011 millimetres squared.

Subtracting these two areas gives the area of the frustum's curved surface at 30,000 millimetres squared.

Lovely stuff, onto the first practise task.

For question one, you'll need to fill in this table of information about the frustum and then use this information to calculate the area of the curved surface of this frustum.

Pause, now to try this question.

For question two, here's another table.

But to fill it in fully, you'll need Pythagoras' theorem.

Pause now to fill in this table and hence find the area of the curved surface of the frustum.

And for question three, calculate the difference in the areas of the curved surfaces of these two frustum.

Pause here for this final question.

Great effort and attention to detail across all of these frustums. For question one, pause here to compare the answers on screen to your own.

The area of the curved surface is 1,210 centimetres squared.

For question two, pause again here to compare answers for a frustum whose curved surface has area of 29,200 millimetres squared.

For question 3A, the radius of the large cone is 105 centimetres and the slant length of the large cone is 165 centimetres.

The curved surface area of this frustum is therefore 53,460 centimetres squared.

On to part B, the slant lengths of the two cones are found using by Pythagoras' theorem at 203 centimetres and 116 centimetres.

The curved surface area of this frustum is therefore 63,137 centimetres squared, which is 9,676 centimetres squared greater than the curved surface area of that other frustum.

So far we've looked quite a lot at just the curved surface of a frustum, but what about its entire surface area? Oh, and also will the definition of surface area change depending on the context that the frustum is used in? Let's have a look.

Okay, here we have Aisha who claims that the curved surface of a frustum is the total surface area of that same frustum.

However Sam disagrees, if we are dealing with a closed frustum, then its total surface area is made up of the curved surface like Aisha says, but the total surface area is also made from the two circular faces on that frustum.

So to find the total surface area of a closed frustum, we need to find the curved surface of that frustum which is found by taking the curved surface of the large cone and subtracting the curved surface of the small cone.

Furthermore however we also need to include the areas of the two circles that are two of the faces of that frustum.

For a frustum, these two circles will always be different in size.

So for this particular closed frustum, we can break down its surface area into three groups of calculations which we will then sum together at the end.

This first group of calculations finds its curved surface which is the subtraction of the curved surfaces of the two cones, the large cone with a radius of 21 and a slant length of 28, and a small cone with a radius of nine and a slant length of 12, giving a total curved surface of 480 pie.

The second group of calculations finds the larger circular face using pi r squared and a radius of 21, giving 441 pi.

The final group of calculations finds the smaller circular face, again using pi r squared and a radius of nine, giving 81 pie.

The total surface area of the whole frustum is the area of the curved surface plus the two circular faces all summed together.

This gives a total surface area of this frustum at 3,150 centimetres squared.

Okay, for this check, we have a large cone that has been split into a smaller cone and a frustum.

Pause here to find the total surface area of this frustum.

We have the difference between the curved surface areas of the two cones, plus the areas of the two circular faces, giving 230 pi.

Next up, here we have a table full of lots of different values on this set of cones and frustum.

Pause here to complete as much of this table as you can.

The curved surfaces of the two cones have areas of 1,440 pi and 40 pi.

Therefore the curved surface of the frustum has an area that is the difference between these two areas at exactly 1,400 pi.

The areas of the two circles are 576 pi and 16 pi, and therefore the total surface area of this whole closed frustum is the sum of these three parts of the frustum at 1,992 pi.

And here for this next check we have a familiar-looking question, but be careful because the information given is slightly different to what you have seen before.

Pause here to first notice all of the details in this question and then use all of these details carefully to then find the total surface area of this frustum.

A very well done if you spotted that the two cones had their total surface area given, not just their curved surface area.

This means for each cone, we first of all need to subtract the area of its circular face which we can find on the frustum.

Now we have only the curved surface area of each cone, we can add on the areas of the two circular faces on the frustum.

Here's an alternate method to get the correct answer if you thought blimey.

I'm subtracting the areas of these circles and then adding them back on.

That's a bit excessive.

Pause here to think about or discuss this alternate method.

Okay, Aisha notices that we can take apart a cone to remove its circular base and then unfold the curved surface which unravels to become a sector of a circle.

Pause here to think about or discuss Aisha's question.

What will an unfolded frustum look like without its two circular faces? Well, let's think about this question with our knowledge of what a frustum is.

A frustum is a cone with a smaller cone removed so its curved surface is going to be a circular sector, but with a smaller circular sector removed.

This C shape is called an annulus, however you do not need to know this term in any detail.

So the surface area of a closed frustum is made of these three two-dimensional shapes.

Let's look in detail first at this annulus.

The inner curved part of the annulus has the same arc length as the circumference of the smaller circle on the frustum, whilst the outer curved part of the annulus has the same arc length as the circumference on the larger circle on the frustum.

Furthermore the radius of the inner sector of the annulus is the slant length of the small cone, whilst the radius of the outer sector of the annulus is the slant length of the large cone.

And finally, one straight edge of the annulus is the slant length of the frustum.

This is because the slant length of the frustum is the difference between the slant lengths of the two cones.

Right, onto the final few checks for understanding.

Here we have a curved surface of a frustum unfolded into this annulus shape.

Pause here to find the lengths labelled a to c.

The outer radius of the annulus is the slant length of the large cone at 60 centimetres.

Similarly the inner radius of the annulus is the slant length of the small cone at 12 centimetres.

The difference between these two radii, the length c, is the slant length of just the frustum.

Now we're given more information.

Pause here once more to complete each of these two sentences.

For the annulus, the inner arc length is the circumference of the smaller circular face of the frustum at 10 pi, whilst the outer arc length is the circumference of the larger circular face of the frustum at 50 pi.

Now for a different annulus unfolded from a frustum, pause here to complete this massive table of information.

We can use the arc lengths of 36 pi and four pi given to find the radii of the two circular faces of the frustum or the two cones.

The curved surface of the large cone uses the radius that we just calculated, as well as the outer radius of the annulus to get 810 pi.

The same is true for the curved surface of the small cone where the inner radius of the annulus is five centimetres, giving us 10 pi.

We subtract these curved surfaces to find the curved surface of the frustum.

The areas of the two circular faces of the frustum use the radii of 18 centimetres and two centimetres to get 18 squared pi and two squared pi.

And as always, the surface area of the closed frustum is the sum of these three parts of the frustum.

Sometimes what is considered surface area will depend on the context that the frustum is used in.

We've looked at closed frustum so far which include both the circular faces.

However some other contexts may only need one circular face and some others may even need neither of the circular faces.

Here is a plant pot that is made of a ratan material and is in the shape of a frustum, but not a fully closed frustum.

We consider its surface area to be the amount of material used to make the plant pot.

Pause here to consider which of these make up the surface area of this plant pot.

The plant pot has a curved surface and a smaller circular face at the bottom of the pot.

There is no larger circular face at the top.

Otherwise it'd be a pretty terrible plant pot.

Brilliant, onto the final few practise questions.

For question one, pause here to complete this table of information and find the surface area of this closed frustum.

For question two, using multiplicative reasoning between the two radii and the two vertical heights and the two slanted heights of a frustum and the two cones, complete this table of information and find the practical surface area of this frustum-shaped bin.

Pause now to dig deep into this tricky question.

And finally, question three.

This annulus is folded up into a lamp shade.

Calculate the amount of material needed to create this lamp shade.

Pause now for question three.

Superb effort on these absolutely challenging questions.

Pause here to compare your answers to question one with those on screen for a closed frustum with a total surface area of 28,900 centimetres squared.

And for question two, the lengths on this large cone are twice the corresponding lengths on the small cone.

We can set up and solve the equation 103 plus h equals 2h to get a vertical height on the small cone at 103 centimetres.

And then we can use Pythagoras' theorem to find the slant length of the small cone at 106 centimetres.

Knowing that the scale factor is two means we can double these lengths to find the vertical height and slant length of the large cone.

These are 206 centimetres and 212 centimetres respectively, but we're not done yet.

The curved surface area of the bin is the difference between the curved surface areas of these two cones at 7,950 pi, but the bin would be absolutely useless without a bottom to keep the rubbish in.

So for its practical surface area, we need to add on a circular face at the bottom of the frustum with a radius of 25 centimetres.

Therefore the total material needed for this bin is 26,900 centimetres squared.

And finally, question three.

On that annulus, the inner radius is 18 centimetres.

Using the two radii of the annulus which are also the slant lengths of the two cones, we can spot that the scale factor is 2.

5.

We are given that 28 pi of metal wire is used around the circumference of the smaller circle on the frustum.

This means the circumference of the larger circle is 70 pi or 28 pi multiplied by the scale factor.

Knowing that the circumference of the large circle is 70 pi, then we know the radius of that circle is 35 centimetres.

The same is true for the small circle whose radius is 14 centimetres from a circumference of 28 pi.

Now we know the radii and slant lengths of both cones, we can calculate the curved surface area of the frustum which is the total surface area of this lampshade which does not have a tangible circular face on either end.

Therefore the surface area of this lampshade is 4,160 centimetres squared.

That was absolutely intense.

An amazing effort, everyone, on this lesson where we have considered the area of the curved surface of a frustum as the difference between the areas of the curved surfaces of the two cones that create that frustum.

If we know the vertical height and radius of a cone, we can find its slant length using pyres theorem.

We do this to help us find the area of the curved surface of a frustum.

For a closed frustum, its surface area is the area of the curved surface plus the area of its two circular faces.

However in a real-world context, not every frustum has two circular faces.

The curved surface of a frustum can be seen as a circular sector with a smaller circular sector removed from it.

You deserve a break after the Herculean effort in today's lesson.

Thank you all so much for your time and focus.

I have been Mr. Gratton and you all have been absolutely amazing.

Take care, everyone, and have an amazing rest of your day.