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I can calculate the volume of a frustum of a cone.

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Licence

This content is © Oak National Academy Limited (2023), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Key learning points

The frustum of a cone can be thought of as a cone with the top missing.
The volume of a frustum of a cone can be thought of as the difference between two cones' volumes.
This formula can be manipulated to rely on information from the frustum only.

Common misconception

Pupils may think that if you cut the height of the cone in a half, the resulting frustrum will have half the volume of the original cone.

The frustrum is actually seven eighths of the volume of the original cone; the small cone that was removed to create the frustrum is one eighth the volume of the original cone if the frustrum and the small cone have the same height.

Keywords

Frustum - A frustum is the 3D shape made from a cone by making a cut parallel to its circular base and removing the resultant smaller cone.
Volume - Volume is the amount of space occupied by a closed 3D shape.

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Hello, I'm Mr. Gratton.

Welcome everyone, and thank you for joining me in this lesson on the volumes of 3D shapes.

Today we will introduce you to what a frustum is, and how to calculate its volume.

Pause here to have a quick look at some definitions that we'll be using today, and we'll be digging deep into what a frustum is later on in the lesson.

First up, let's look at some frustums in comparison to two similar cones.

Aisha here wants to cut this large cone to make a smaller cone, and Sam spots that the shape left over is pretty unusual.

Does it have a name? Well, it does.

It is called a frustum.

Not a \"frosterum\", as some people accidentally call it.

A frustum is a three-dimensional shape made from making a circular cut from a large cone that is parallel to its circular base.

Doing this cut will split the large cone into two different three dimensional shapes, a smaller cone and then the frustum.

For this quick check, pause here to identify which of these cuts will result in a frustum.

Only B and D will.

A is incorrect, because that cut is not parallel to the base of the cone.

Whilst in C, that cut looks perpendicular to the base, not parallel to it.

The large and small cones both have different radii on their circular faces.

To identify them, we call the radius of the large cone capital R, and the radius of the small cone lowercase r.

The frustum has both of these circular faces, one with each radius, capital R and lowercase r.

Also, a line segment that passes through both the centres of both circular faces of the frustum is always perpendicular to both these faces.

Things are very similar for the height of the two cones.

Both the large cone and the small cone have different heights, where the height of the large cone is capital H, and the height of the small cone is lowercase h.

The height of the frustum is the difference between these two heights, and therefore the expression for the height of the frustum is H - h.

Okay, on the left we have a large cone, and then in the middle and on the right, we have a small cone cut from the large cone, and the resultant frustum.

Pause here to find the lengths, labelled a to c.

A is the radius at the base of the large cone, at 12 centimetres.

B is the radius of the cut, the base of the small cone, at three centimetres.

C is the difference between the heights of the two cones, at 21 centimetres.

The small cone that is removed to make a frustum can be shown with dotted lines like this.

This helps us to see the height of both the original large cone, and the small cone that was removed.

In order to calculate other properties of the frustum, such as its volume.

We can calculate the volume of a frustum by finding the difference in the volumes of the big cone and the small cone.

For example, let's look at this frustum.

The volume of a cone is 1/3 multiplied by pi r-squared h, where r is the radius of the cone, and h is the height of the cone.

The goal is to find both the radii and the heights of both the large and the small cones, so that we can calculate the volumes of each cone using this formula.

Currently, we know both radii, that's six centimetres and nine centimetres, and we know the height of the large cone, but we do not know the height of the small cone, only the frustum.

The height of the small cone is 20 centimetres, which is the difference between the height of the large cone and the frustum.

So, let's take all of this information and substitute it into one of two copies of this volume formula.

With this first one showing the volume of the larger cone at 810 pi centimetres cubed.

And the second version showing the volume of the smaller cone, at 240 pi centimetres cubed.

The volume of the frustum is the difference between these two volumes.

At 810 pi, subtract 240 pi, or 570 pi centimetres cubed.

For this check, pause here to match the values shown at the bottom of this slide to the features of the frustum or cone.

The height of the small cone is the difference between the heights of the large cone and the frustum.

The volume of the small cone is 567 pi, whilst the volume of the large cone is shown in two different ways.

Well done if you spotted both of these ways.

The volume of the frustum is the difference between 2625 pi and 567 pi, giving 2058 pi.

And for this next check, pause here.

Define the volume of the small cone and the frustum in order to show that Sam's observation is not correct.

The volume of the small cone is 1140 centimetres cubed, whilst the volume of the frustum is 7980 centimetres cubed.

We can see here that the frustum is not equal in volume to the small cone.

In fact, the frustum has a volume that is seven times the volume of the small cone.

Lovely stuff.

Let's practise our understanding of frustums. For question one, find the lengths labelled on the large cone and the frustum, whilst for question two, complete these calculations in order to find the volume of the two cones and the frustum.

Pause now for these two questions.

And for question three, pause here to calculate the volume of each frustum.

Hey, great effort on this practise task! Onto the answers for question one.

A centimetres is two centimetres, b centimetres is five centimetres, and c centimetres is 20 centimetres.

For question two, the volumes of the large and small cones were 1458 pi, and two pi, meaning that the volume of the frustum itself is the difference between these two other volumes at 1456 pi.

Onto question three, the volumes of the two cones for part A are 416,745 pi, and 184,320 pi.

Meaning that the volume of the frustum itself is 730,000 cubic millimetres.

For part B, the radius at the top of the frustum is 54 millimetres, and the radius at the bottom of the frustum is 135 millimetres.

The frustum itself has a volume of 1,970,000 cubic millimetres.

And for part C, the radius at the top of the frustum is 35 centimetres, and the radius at the bottom of the frustum is 15 centimetres, meaning that the frustum itself has a volume of 223,000 cubic centimetres.

Now that we've seen frustums shown in their 3D form, let's analyse frustums from their front elevation, side elevation, and plan viewpoints.

But first, let's recap what these viewpoints even are.

We have the front elevation and side elevation of an object, and we also have the plan view, which is the viewpoint from directly above the object.

For a frustum or for a cone, the front and side elevations will look exactly the same.

Any frustum will look like an isosceles trapezium from its front or side elevation, like this.

But why? Well, this is because both the large cone and the small cone will look like isosceles triangles, like this.

And when you remove a smaller isosceles triangle from a larger one, you're left with the isosceles trapezium at the bottom of that triangle.

Next up, from the top, the plan view.

For any frustum, we will see two circles, one for each of its two circular faces.

The shared centre of the two circles aligned with the apex of the original cone in this plan view of a right circular cone.

For this check, we're looking at the front elevation for some different 3D shapes.

Pause here to identify the 3D shape that is not a frustum.

A is the viewpoint from a frustum, with the dotted line segments to represent the small cone that was removed.

B is the viewpoint of only the frustum.

However, c is not an isosceles trapezium, and so it cannot be a viewpoint from a frustum.

Next up, the plan view.

Pause here to identify the non-frustums from this different viewpoint.

A is a frustum, one with a large circular face, and a much smaller second circular face.

This means that the cone that was removed was a very small part of the original larger cone.

B is not a frustum, since the two circles do not share a centre.

And whilst c looks suspiciously close to a frustum, the smaller shape is not a circle, rather a more general ellipse.

We can identify the heights and radii from a frustum either from its 3D representation, or from its different 2D viewpoints.

For example, the height of the frustum is also the height of the trapezium from its front elevation.

For this particular frustum, the height is three units, and the height of the small cone is nine units.

Meaning that the height of the large cone is 3 + 9, 12 units.

Similarly, the radii of the two circular faces on the frustum are also the radii of the two circles on the plan view.

The radius of the larger circle is the radius of the larger circular face, and the same is true for the smaller circle on the plan view, when compared to the smaller circular face.

However, there are actually two different ways of spotting the two radii.

We can also find it on the front or side elevations.

We identify the two parallel sides.

Those two parallel sides are the diameters of the two circular faces of the frustum.

And so, half the length of these two parallel sides are the radii of those two circular faces.

Half the length of the shorter parallel side is the smaller radius, and half the length of the longer parallel side is the larger radius.

We can calculate the volume of a frustum using only its plan and elevations.

This is because we only need to find the heights and radii of both the large and small cones.

For the large cone, the radius is four centimetres, found from any of the viewpoints, and the height is 12 units found from the front or side elevation.

The volume of the large cone is 64 pi.

For the small cone, the radius is three units, and the height is nine units, giving a volume of 27 pi.

And so, just like always, the frustum's volume is the difference between the volumes of the two cones, at 37 pi cubic units.

Right.

Onto a check.

Pause here to complete each of these sentences.

The height of the frustum is six units.

The radius of the larger circular face on the frustum, also the circular face on the larger cone, is five units.

The height of the small cone removed to make the frustum is four units.

Okay, great stuff.

Let's put your understanding of these viewpoints into practise.

Pause here for question one.

Fill in the table from the front elevation and plan view given.

For question two, here's a side elevation of a frustum.

Find the volume of that frustum.

And for question three, you are only given the plan view of a frustum, but you are given the heights of the small cone and frustum.

Find the volume of that frustum.

Pause here for these two questions.

Great effort on all three questions.

Pause here to check your answers to question one for a frustum with volume of 84 pi cubic units.

And for question two, the large cone has a radius of 33 centimetres.

Do not let the fact we are given a diameter catch you out.

The volume of the frustum is 19,656 pi cubic centimetres.

And for question three, the large cone has a height of 63 inches, which is the sum of the heights of the frustum and small cone.

The radius of the large cone is the sum of 20 inches, which is the radius of the small cone, and 15 inches, which is the length between those two circles.

The volume of the frustum is therefore 66,000 cubic inches.

And lastly, there is more to the relationship between the lengths on the large cone, the small cone, and the frustum, than we have currently seen.

Let's use some of these relationships to help us find the volume of a frustum when given less information than before.

For example, Sam seems to think that we can't find the volume of this particular frustum.

This is because we need to know the height of that small cone, which we currently have absolutely no way of calculating.

However, Aisha holds some hope.

Since we know both the radii on both circular faces on the frustum, can we use these lengths to our advantage to calculate the heights? Well, let's have a look.

In fact, let's go back to the beginning of this lesson.

Here is a large cone and a small cone, cut from the large one, in order to make a frustum.

Both the large cone and small cone are actually similar to each other.

Because there is similarity, both radii and both heights share the same multiplicative relationship.

We can use this multiplicative relationship between the radii in order to find the heights of both cones, as long as we are given the height of the frustum to start with.

Right.

Let's have a look at this frustum and pair of cones.

Because there's a multiplicative relationship, we can represent the radii and heights on a ratio table.

This ratio table will compare the small and large cones.

So let's fill in what we know.

This pair of radii has a multiplicative relationship of three, meaning that the height of the large cone will also be three times the height of the small cone.

Right now, we do not know the height of the small cone, so let's call it h.

And so the height of the large cone is 3h.

But this alone does not help us.

There is a second algebraic representation for the height of the large cone.

The height of the large cone is also H + 20.

This is because H + 20 is the sum of the heights of the small cone and frustum.

Now we have two equal representations for the height of the large cone, we can set up an equation between both these representations.

3h from the ratio table equals h + 20 from the sum of the height of the small cone and frustum.

Solving this equation gives h=10.

H is the height of the small cone, meaning that the height of the large cone is three lots of 10, at 30.

Brilliant.

Now we know both radii and both heights, we can find the volume of the frustum.

The volume of the large cone subtract the volume of the small cone, gives the volume of the frustum, at 3120 pi centimetres cubed.

For this final pair of checks, pause here to complete this ratio table.

A and b are numerical values, whilst c and d are algebraic expressions.

The two radii are 16 and 20, and so there is a multiplicative relationship between the small and large cone of 5/4.

The height of the small cone is h, and so the height of the large cone is 5/4 h.

Now using this ratio table, pause here to construct a linear equation and find the height of the large cone.

This is the equation.

1/4 h = 12, meaning the h itself equals 48.

The height of the large cone is 48 + 12, the height of the frustum, at 60.

Alternatively, 5/4 of 48 will also give you 60.

The height of the large cone is 60 centimetres.

Amazing.

Onto the final few practise questions.

For question one, pause here to complete this ratio table, construct a linear equation, and then use all of the information that you know to find the volume of this frustum.

For question two, find the volume of the frustum, and drawing a ratio table will help.

And the final question, question three.

Here is a sketch of the side elevation of a frustum.

Using this sketch, calculate the volume of the frustum.

Pause now for these final two questions.

Great stuff.

Onto the answers.

For question one, the ratio table will have 2 and 16 as the two radii, and h and 8h as the two heights.

8h = h + 42, giving h = 6.

We can use this information to find the volumes of the small and large cones, giving the volume of the frustum as 4,096 pi, take away 8 pi, which equals 12,800 centimetres cubed for the volume of the frustum, rounded to three significant figures.

For question two, pause here to compare your calculations for the frustum to those on screen.

The frustum itself has a volume of 2.

75 million centimetres cubed.

For question three, we first need to find the height of the frustum using the area of a trapezium.

The height of the frustum is 44 centimetres.

We can then construct a ratio table to calculate the heights of the small and large cones, as 11 centimetres and 55 centimetres.

Using the heights we calculated, we can find the volumes of the small and large cones at 33 pi and 4,125 pi, meaning that the frustum has a volume of 12,900 centimetres cubed.

Superb effort, everyone, in the learning and adapting to a new 3D shape, in a lesson where we have seen that a frustum is a 3D object, made from cutting a smaller similar cone from an original larger one, where the volume of a frustum is the difference between the volumes of the smaller and larger cones.

We can find the radii and heights of these cones, and resultant frustum, from the front and side elevations of these shapes, as well as their plan view.

And finally, multiplicative reasoning comes to the rescue once more.

We can find the heights of either the larger or smaller cones, or of the frustum, through the shared multiplicative relationship with the two radii.

Once again, a massive \"well done\" for your effort during this lesson.

I've been Mr. Gratton, and you have been absolutely spectacular! Until next time, everyone.

Take care, and goodbye.

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Give your answer to the nearest cubic centimetre.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719415153%2Fhxkrtjkvwr1haxz9mptg.jpg&w=640&q=75", "imageKey": "9ecf6a14-bace-b543-4954-1cccb3273086/question/a8eb774e3ca824caf295c6d3381cc5a8.jpeg" }, "questionType": "Multiple choice", "correct": [ { "text": "862 cm³" } ], "choices": [ { "text": "652 cm³" }, { "text": "862 cm³" }, { "text": "1124 cm³ " }, { "text": "2694 cm³" } ], "id": "59216c74-abd6-8cd6-abbc-8c2db72492d2", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" }, { "type": "Question", "question": { "text": "This composite solid is constructed with a cylinder and a hemisphere. Each have a diameter of 12 cm. The volume of the solid is ______cm³ (correct to 4 significant figures).", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719415154%2Fhanb2eiklqbpct1wjdkx.jpg&w=640&q=75", "imageKey": "9ecf6a14-bace-b543-4954-1cccb3273086/question/0feb0f2fdd8891d2f8f7ceca3b804cda.jpeg" }, "questionType": "Short answer", "correct": [ { "text": "2149" } ], "id": "b699c305-e8c3-ed22-17e1-d83eeca0f520", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" }, { "type": "Question", "question": { "text": "This composite solid is constructed with a cone and a hemisphere. The volume of the solid, in terms of 𝜋, is ______𝜋 cm³.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719568550%2Fgklsuj3gw5ffw3sg2yng.jpg&w=640&q=75", "imageKey": "9ecf6a14-bace-b543-4954-1cccb3273086/question/642b4968ffb6daffe1645ecad3cd14cc.jpeg" }, "questionType": "Short answer", "correct": [ { "text": "264" } ], "id": "51b09190-9a50-0682-2613-fa768e00cef0", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" } ], "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea", "meta": { "gradeStageId": "ks4", "subject": "Maths", "tier": "Higher", "grade": "Year 11", "topic": null, "unit": "2D and 3D shape: surface area and volume (pyramids, spheres and cones)", "lessonCount": "18", "age": [ 15, 16 ], "gradeId": "age_15_16", "gradeStage": "GCSE", "lesson": "Volume of a frustum of a cone" } }, { "id": "d81e9b9e-25b7-efde-5920-4b4f0a56c036", "type": "Resource", "resourceType": "Quiz", "title": "Volume of a frustum of a cone", "quizType": "Exit quiz", "items": [ { "type": "Question", "question": { "text": "A ______ is the 3D shape made from a cone by making a cut parallel to its circular base and removing the resultant smaller cone." }, "questionType": "Short answer", "correct": [ { "text": "frustum" } ], "id": "bd76b53e-8cd4-629f-5631-4b22c1a1fcc5", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" }, { "type": "Question", "question": { "text": "The left-hand cone is split into a smaller cone and a frustum. Match each length labelled a - c to its value.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719572803%2Fjh00tjzgbyuwrv7mle5k.jpg&w=640&q=75", "imageKey": "d81e9b9e-25b7-efde-5920-4b4f0a56c036/question/f175296e9f3cc9a0d3ad9e8cc308d29d.jpeg" }, "questionType": "Matching", "pairs": [ { "left": { "text": "a" }, "right": { "text": "15 cm" } }, { "left": { "text": "b" }, "right": { "text": "5 cm" } }, { "left": { "text": "c" }, "right": { "text": "20 cm" } } ], "id": "9eba6399-b7b9-8e77-f8fd-30853b8e2e92", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" }, { "type": "Question", "question": { "text": "The diagram shows a side elevation of a frustum. Find the volume of the large cone that the frustum is cut from. Give your answer correct to 3 significant figures.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719572804%2Fkundfvgw5uowzrpjjxlx.jpg&w=640&q=75", "imageKey": "d81e9b9e-25b7-efde-5920-4b4f0a56c036/question/c93bea7107b6acecad71591535de3520.jpeg" }, "questionType": "Multiple choice", "correct": [ { "text": "12 500 cm³ to 3 s.f." } ], "choices": [ { "text": "3421 cm³ to 3 s.f." }, { "text": "9120 cm³ to 3 s.f." }, { "text": "12 500 cm³ to 3 s.f." }, { "text": "50 200 cm³ to 3 s.f." } ], "id": "0382498d-50f3-468c-67c2-72bfa6673a07", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" }, { "type": "Question", "question": { "text": "The diagram shows a side elevation of a frustum. Find the volume of the frustum. Give your answer correct to 3 significant figures.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719572804%2Fkundfvgw5uowzrpjjxlx.jpg&w=640&q=75", "imageKey": "d81e9b9e-25b7-efde-5920-4b4f0a56c036/question/c93bea7107b6acecad71591535de3520.jpeg" }, "questionType": "Multiple choice", "correct": [ { "text": "7720 cm³" } ], "choices": [ { "text": "4294 cm³" }, { "text": "7720 cm³" }, { "text": "8790 cm³" }, { "text": "12 500 cm³" } ], "id": "08ef4607-e55e-ceba-e3ea-ef9bf6efa322", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" }, { "type": "Question", "question": { "text": "A cone is cut into a frustum of height 20 cm and a small cone. The radius of the base of the frustum is 15 cm and the radius of its top is 9 cm. The height of the small cone is ______ cm.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719572804%2Fip4xp1yyrcnxn1keacft.jpg&w=640&q=75", "imageKey": "d81e9b9e-25b7-efde-5920-4b4f0a56c036/question/796b0cac157e87ecc931bbaa7972cae7.jpeg" }, "questionType": "Short answer", "correct": [ { "text": "30" } ], "id": "8f6964d3-898d-cb65-9ec0-3ef87dcb3da1", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" }, { "type": "Question", "question": { "text": "A cone is cut into a frustum of height 20 cm and a small cone. The radius of the base of the frustum is 15 cm and the radius of its top is 9 cm. The volume of the frustum is ______𝜋 cm", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1719572804%2Fip4xp1yyrcnxn1keacft.jpg&w=640&q=75", "imageKey": "d81e9b9e-25b7-efde-5920-4b4f0a56c036/question/796b0cac157e87ecc931bbaa7972cae7.jpeg" }, "questionType": "Short answer", "correct": [ { "text": "2940" }, { "text": "2 940" }, { "text": "2,940" } ], "id": "538a8df4-d346-3038-29ea-86ba13208b5c", "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea" } ], "lessonId": "43c89cd5-1f0d-4213-3328-aff9025a8aea", "meta": { "gradeStageId": "ks4", "subject": "Maths", "tier": "Higher", "grade": "Year 11", "topic": null, "unit": "2D and 3D shape: surface area and volume (pyramids, spheres and cones)", "lessonCount": "18", "age": [ 15, 16 ], "gradeId": "age_15_16", "gradeStage": "GCSE", "lesson": "Volume of a frustum of a cone" } } ] }