AFCAT Numerical Ability · Surface Area and Volume

Cone and Frustum

Slant height, curved and total surface and volume of a cone, a cone cut parallel to its base, and the frustum that is left.

All Surface Area and Volume notes

Why this matters

Nearly every cone question needs the slant height first, from l² = r² + h², and the common slip is using the height where the slant belongs. The cut-cone and frustum questions look heavy but rest on one fact: the small cone on top is a scaled copy of the whole, so its lengths, areas and volumes follow k, k² and k³.

Concept 1 of 4: Slant height and the surface of a cone

Cut a cone straight down through its tip and you see a triangle. Half of it is a right triangle with the radius and the height as its legs and the slant height as its long side. The curved surface is the slant height times half the base's circumference.

Definition

Slant height $l = \sqrt{r^2 + h^2}$ . Useful triples: 3, 4, 5; 5, 12, 13; 8, 15, 17; 9, 40, 41.
Curved surface $\pi rl$ ; total surface $\pi r(l + r)$ , which adds the base circle.
A conical tent or a funnel is the curved surface only. Cloth of a given width: length = area ÷ width.

Cone of radius r, height h, slant l

l = \sqrt{r^2 + h^2},\quad \text{CSA} = \pi r l,\quad \text{TSA} = \pi r (l + r)

Worked example

A cone has radius 9 cm and height 12 cm. Find its slant height, curved surface and total surface, in terms of

\pi

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The same idea in a practice question:

Practice questionEasy

Example 1 · Surface Area and Volume · Cone and Frustum

How many metres of cloth 50 m wide will be required to make a conical tent, the radius of whose base is 7 m and whose height is 24 m?

Curved surface uses the slant height

For radius 9 and height 12,

\pi rh = 108\pi

is not a surface of anything. The curved surface is

\pi rl = 135\pi

Cloth length is area divided by width

600 m² of canvas from cloth 3 m wide needs 200 m of cloth. Stopping at the area answers a different question.

Drill 5 more on slant height and the surface of a cone

Concept 2 of 4: Volume of a cone

A cone holds a third of the cylinder on the same base and height, just as a pyramid holds a third of its prism. When radius and height come as a ratio, write them as multiples of one number k.

Definition

Volume $\tfrac 13 \pi r^2 h$ .
Radius and height in the ratio $p : q$ : take $r = pk$ , $h = qk$ . The volume gives $k^3$ ; the slant height is then $k\sqrt{p^2 + q^2}$ .
Volume goes as $r^2 h$ : height doubled doubles it, radius doubled makes it 4 times, both doubled 8 times.

Cone

V = \tfrac13 \pi r^2 h

Worked example

Find the volume of a cone of radius 21 cm and height 10 cm. (Use

\pi = \tfrac{22}{7}

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The same idea in a practice question:

Practice questionModerate

Example 2 · Surface Area and Volume · Cone and Frustum

The radius and height of a right circular cone are in the ratio of 5 : 12 and its volume is

2512 \text{ cm}^3

. The slant height of the cone is

Doubling the height is a 100% rise

Volume is proportional to height, so twice the height is twice the volume: an increase of 100%, not 200%.

The radius counts squared

Doubling only the radius makes the cone hold 4 times as much. Treating the radius like the height gives 2 times.

Drill 4 more on volume of a cone

Concept 3 of 4: A cone cut parallel to its base

A cut parallel to the base takes a small cone off the top, and that small cone is the same shape as the whole. If its height is a fraction k of the full height, so are its radius and slant height; its surface is k² of the whole and its volume k³.

Definition

Scale $k = \dfrac{\text{small cone's height}}{\text{whole height}}$ , the same for radius and slant height.
Small cone: curved surface $k^2$ and volume $k^3$ of the whole cone.
The piece left below (the frustum) has volume $1 - k^3$ and curved surface $1 - k^2$ of the whole.
Heights 'above the base' are measured from the base; the small cone's height is measured down from the tip.

Small cone cut off the top

\dfrac{V_{\text{top}}}{V} = k^3,\quad \dfrac{\text{CSA}_{\text{top}}}{\text{CSA}} = k^2

Worked example

A cone 16 cm tall with base radius 8 cm is cut parallel to its base 4 cm below the tip. Find the radius of the cut, and what fraction of the volume and of the curved surface the frustum keeps.

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The same idea in a practice question:

Practice questionModerate

Example 3 · Surface Area and Volume · Cone and Frustum

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be

\frac{1}{27}

of the volume of the given cone, then the height above the base where the section is made, is

Measure from the right end

If the small cone is 9 cm of a 45 cm cone, the cut is 36 cm above the base. The question may ask for either number.

Halving the height does not halve the volume

A cut through the mid-point leaves a top cone with

\tfrac 18

of the volume and a frustum with

\tfrac 78

Drill 4 more on a cone cut parallel to its base

Concept 4 of 4: The frustum: a bucket shape

A frustum is what is left of a cone when the top is cut off: two circles of radii R and r, a height h between them. A bucket is one. Its slant height comes from a right triangle whose legs are the height and the difference of the radii.

Definition

Slant height $l = \sqrt{h^2 + (R - r)^2}$ .
Curved surface $\pi(R + r)l$ ; total surface adds both circles, $\pi R^2 + \pi r^2$ . An open bucket adds only its bottom.
Volume $\dfrac{\pi h}{3}(R^2 + r^2 + Rr)$ .
The cone it came from: the cut-off cone of height $t$ satisfies $\dfrac{t + h}{t} = \dfrac{R}{r}$ .

Frustum of radii R and r, height h

V = \dfrac{\pi h}{3}(R^2 + r^2 + Rr),\quad l = \sqrt{h^2 + (R - r)^2},\quad \text{CSA} = \pi (R + r) l

Worked example

A frustum has end radii 11 cm and 5 cm and height 8 cm. Find its slant height, total surface and volume, in terms of

\pi

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The same idea in a practice question:

Practice questionModerate

Example 4 · Surface Area and Volume · Cone and Frustum

The radii of the circular ends of a bucket of height 40 cm are of lengths 35 cm and 14 cm. What is the volume of the bucket?

The slant uses R − r, not R + r

Radii 11 and 5 with height 8 give slant

\sqrt{64 + 36} = 10

. Using

R + r = 16

gives

\sqrt{320}

, which is wrong.

An open bucket has one circle

The metal in a bucket with no lid is its curved surface plus the small bottom circle only.

Drill 3 more on the frustum: a bucket shape

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Slant height and the surface of a cone
Cone of radius r, height h, slant l
$l = \sqrt{r^2 + h^2},\quad \text{CSA} = \pi r l,\quad \text{TSA} = \pi r (l + r)$
Volume of a cone
Cone
$V = \tfrac13 \pi r^2 h$
A cone cut parallel to its base
Small cone cut off the top
$\dfrac{V_{\text{top}}}{V} = k^3,\quad \dfrac{\text{CSA}_{\text{top}}}{\text{CSA}} = k^2$
The frustum: a bucket shape
Frustum of radii R and r, height h
$V = \dfrac{\pi h}{3}(R^2 + r^2 + Rr),\quad l = \sqrt{h^2 + (R - r)^2},\quad \text{CSA} = \pi (R + r) l$

Watch out for (8)

Curved surface uses the slant height
→ Slant height and the surface of a cone
Cloth length is area divided by width
→ Slant height and the surface of a cone
Doubling the height is a 100% rise
→ Volume of a cone
The radius counts squared
→ Volume of a cone
Measure from the right end
→ A cone cut parallel to its base
Halving the height does not halve the volume
→ A cone cut parallel to its base
The slant uses R − r, not R + r
→ The frustum: a bucket shape
An open bucket has one circle
→ The frustum: a bucket shape

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