Surface Area of a Cone

Find the curved surface area of a cone, $\pi r l$ , and its total surface area, $\pi r l + \pi r^{2}$ , finding the slant height $l$ from the radius and the height where you need it.

Before you start

A circle of radius $r$ has circumference $2\pi r$ and area $\pi r^{2}$ .
A cone has a radius $r$ , a height $h$ and a slant height $l$ , with $l^{2} = r^{2} + h^{2}$ (Volume of a Cone).

The Net of a Cone

A cone has two surfaces: the flat circular base and the curved surface. Cut the curved surface along a slant height and lay it flat, and it becomes a sector of a circle, a shape like a slice of a round cake.

The radius of the sector is the slant height $l$ of the cone, because every point of the base circle is $l$ from the apex. The curved edge of the sector (its arc) wraps once round the base, so it is as long as the base circle: $2\pi r$ .

The curved surface opens out into this sector of radius $l$ , and its area works out to $\pi r l$ , quoted here without its proof.

Surface area of a cone

Curved surface area $= \pi r l$

Total surface area of a solid cone $= \pi r l + \pi r^{2}$ (the curved surface and the base)

Here $r$ is the radius of the base and $l$ the slant height, not the height.

Example 1: Radius and slant height

A solid cone has a base radius of $5$ cm and a slant height of $13$ cm. Find (a) its curved surface area, (b) its total surface area.

Solution

(a) Curved surface $= \pi \times 5 \times 13 = 65\pi = 204.20\ldots = 204$ (3 s.f.)

(b) Base $= \pi \times 5^{2} = 25\pi$ . Total $= 65\pi + 25\pi = 90\pi = 282.74\ldots = 283$ (3 s.f.)

Add the exact amounts ( $65\pi + 25\pi$ ) rather than the rounded ones.

(a) $204 \text{ cm}^{2}$ (b) $283 \text{ cm}^{2}$

Example 2: In terms of π

A solid cone has a radius of $9$ cm and a slant height of $15$ cm. Find its total surface area, leaving your answer in terms of $\pi$ .

Solution

Curved surface $= \pi \times 9 \times 15 = 135\pi\text{.}$ Base $= \pi \times 9^{2} = 81\pi$ .

Total $= 135\pi + 81\pi = 216\pi$

$216\pi \text{ cm}^{2}$

Example 3: Taking π = 3.142

A cone has a base of diameter $14$ cm and a slant height of $20$ cm. Taking $\pi = 3.142$ , find its curved surface area.

Solution

$r = 14 \div 2 = 7$ cm.

Curved surface $= 3.142 \times 7 \times 20 = 439.88 = 440$ (3 s.f.)

$440 \text{ cm}^{2}$

When the Height Is Given

The curved surface needs the slant height. If a question gives the height instead, find $l$ first from $l^{2} = r^{2} + h^{2}$ .

Example 4: Radius and height

A solid cone has a base radius of $6$ cm and a height of $8$ cm. Find its total surface area.

Solution

$l^{2} = 6^{2} + 8^{2} = 36 + 64 = 100\text{,}$ so $l = 10$ cm.

Curved surface $= \pi \times 6 \times 10 = 60\pi\text{.}$ Base $= \pi \times 6^{2} = 36\pi$ .

Total $= 96\pi = 301.59\ldots = 302$ (3 s.f.)

The total surface area is $302 \text{ cm}^{2}$ .

Working Back from an Area

Example 5: The slant height and the height

The curved surface area of a cone is $150\pi \text{ cm}^{2}$ and its radius is $6$ cm. Find (a) its slant height, (b) its height.

Solution

(a) $\pi \times 6 \times l = 150\pi$ , so $6l = 150$ and $l = 25$ cm.

(b) $h^{2} = l^{2} - r^{2} = 625 - 36 = 589\text{,}$ so $h = \sqrt{589} = 24.26\ldots = 24.3$ (3 s.f.)

(a) $25$ cm (b) $24.3$ cm

Example 6: The radius

A cone has a slant height of $12$ cm and a curved surface area of $200 \text{ cm}^{2}$ . Find its base radius.

Solution

$\pi \times r \times 12 = 200$

$r = \frac{200}{12\pi} = 5.305\ldots = 5.31$ (3 s.f.)

The radius is $5.31$ cm.

Hats, Cups and Roofs: Cones without a Base

A party hat, a paper cup, an ice-cream wafer or a cone-shaped roof has no base: only its curved surface is made. Use $\pi r l$ alone, and say which you mean: a hollow cone open at its base needs only the curved surface; a solid cone has the base as well.

Example 7: A party hat

A party hat is a cone, open at the base, with a radius of $8$ cm and a height of $15$ cm. Find the area of card needed to make it.

Solution

$l^{2} = 8^{2} + 15^{2} = 64 + 225 = 289\text{,}$ so $l = 17$ cm.

Card (curved surface only) $= \pi \times 8 \times 17 = 136\pi = 427.25\ldots = 427$ (3 s.f.)

About $427 \text{ cm}^{2}$ of card

Example 8: Painting a roof

The roof of a round pavilion is a cone with a radius of $3$ m and a height of $4$ m. One litre of paint covers $6 \text{ m}^{2}$ . How many whole litres must be bought to paint the roof once?

Solution

$l^{2} = 3^{2} + 4^{2} = 25$ , so $l = 5$ m.

Roof (curved surface) $= \pi \times 3 \times 5 = 15\pi = 47.12\ldots \text{ m}^{2}$

Paint: $47.12\ldots \div 6 = 7.85\ldots$ litres. $7$ litres would not be enough, so round up.

$8$ litres

Common Mistakes

Using the height for $l$ . For radius $6$ cm and height $8$ cm, $\pi \times 6 \times 8$ is wrong; the slant height is $10$ cm, so the curved surface is $60\pi$ .
Adding a base that is not there, for a hat, a cup or a roof, or leaving it out of a solid cone.
Using the diameter for $r$ .
Mixing the formulas. $\pi r l$ is an area (two lengths multiplied); $\frac{1}{3}\pi r^{2} h$ is a volume (three lengths).
Rounding $l$ before the last step. Keep $\sqrt{589}$ (Example 5) in the calculator.

Practice Questions

Try each question before you open its solution. Use the calculator's $\pi$ unless the question says otherwise, and give answers correct to 3 significant figures where they are not exact.

Question 1

Find the curved surface area of a cone with a radius of $4$ cm and a slant height of $9$ cm.

ShowHide solution

$\pi \times 4 \times 9 = 36\pi = 113.09\ldots = 113$ (3 s.f.)

$113 \text{ cm}^{2}$

Question 2

Find the total surface area of a solid cone with a radius of $7$ cm and a slant height of $10$ cm.

ShowHide solution

$\pi \times 7 \times 10 + \pi \times 7^{2} = 70\pi + 49\pi = 119\pi = 373.84\ldots = 374$ (3 s.f.)

$374 \text{ cm}^{2}$

Question 3

A solid cone has a radius of $12$ cm and a height of $16$ cm. Find (a) its slant height, (b) its curved surface area, (c) its total surface area.

ShowHide solution

(a) $l^{2} = 144 + 256 = 400\text{,}$ so $l = 20$ cm.

(b) $\pi \times 12 \times 20 = 240\pi = 753.98\ldots = 754$ (3 s.f.)

(a) $20$ cm (b) $754 \text{ cm}^{2}$ (c) $1210 \text{ cm}^{2}$

Question 4

A solid cone has a radius of $3$ cm and a height of $4$ cm. Find its total surface area in terms of $\pi$ .

ShowHide solution

$l = \sqrt{9 + 16} = 5$ cm. $\pi \times 3 \times 5 + \pi \times 3^{2} = 15\pi + 9\pi = 24\pi$

$24\pi \text{ cm}^{2}$

Question 5

A solid cone has a base of diameter $10$ cm and a slant height of $12$ cm. Taking $\pi = 3.142$ , find its total surface area.

ShowHide solution

$r = 5$ cm. Curved: $3.142 \times 5 \times 12 = 188.52\text{.}$ Base: $3.142 \times 25 = 78.55$ .

Total: $188.52 + 78.55 = 267.07 = 267$ (3 s.f.)

$267 \text{ cm}^{2}$

Question 6

A cone has a slant height of $11$ cm and a curved surface area of $88\pi \text{ cm}^{2}$ . Find (a) its radius, (b) its height.

ShowHide solution

(a) $11\pi r = 88\pi$ , so $r = 8$ cm.

(b) $h^{2} = 11^{2} - 8^{2} = 121 - 64 = 57\text{,}$ so $h = \sqrt{57} = 7.549\ldots = 7.55$ (3 s.f.)

(a) $8$ cm (b) $7.55$ cm

Question 7

A solid cone of radius $15$ cm has a total surface area of $600\pi \text{ cm}^{2}$ . Find its height.

ShowHide solution

$15\pi l + 225\pi = 600\pi\text{,}$ so $15l = 375$ and $l = 25$ cm.

$h^{2} = 25^{2} - 15^{2} = 625 - 225 = 400\text{,}$ so $h = 20$ cm.

$20$ cm

Question 8

An ice-cream wafer is a cone, open at the top, with a radius of $3$ cm and a depth of $10$ cm. Find the area of wafer.

ShowHide solution

$l^{2} = 3^{2} + 10^{2} = 109$ , so $l = \sqrt{109} = 10.44\ldots$ cm.

Curved surface only: $\pi \times 3 \times \sqrt{109} = 98.39\ldots = 98.4$ (3 s.f.)

$98.4 \text{ cm}^{2}$

Question 9

A conical tent with no floor has a radius of $2.5$ m and a height of $1.8$ m. Find the area of canvas needed.

ShowHide solution

$l^{2} = 2.5^{2} + 1.8^{2} = 6.25 + 3.24 = 9.49\text{,}$ so $l = 3.080\ldots$ m.

Canvas $= \pi \times 2.5 \times 3.080\ldots = 24.19\ldots = 24.2$ (3 s.f.)

$24.2 \text{ m}^{2}$

Question 10

Siti finds the curved surface area of a cone of radius $7$ cm and height $24$ cm as $\pi \times 7 \times 24 = 528 \text{ cm}^{2}\text{.}$ Explain her mistake and find the correct answer.

ShowHide solution

She used the height, $24$ cm, but the curved surface needs the slant height.

$l = \sqrt{7^{2} + 24^{2}} = \sqrt{625} = 25$ cm, so the curved surface is $\pi \times 7 \times 25 = 175\pi = 549.77\ldots = 550$ (3 s.f.)

She used $h$ for $l$ . The curved surface area is $550 \text{ cm}^{2}$ .

Question 11

The sector below, of radius $20$ cm, is rolled up to make the curved surface of a cone with a base radius of $7$ cm. Find (a) the area of the sector, (b) the height of the cone.

ShowHide solution

(a) The sector is the curved surface, with $l = 20$ cm and $r = 7$ cm: $\pi \times 7 \times 20 = 140\pi = 439.82\ldots = 440$ (3 s.f.)

(b) $h^{2} = 20^{2} - 7^{2} = 400 - 49 = 351\text{,}$ so $h = \sqrt{351} = 18.73\ldots = 18.7$ (3 s.f.)

(a) $440 \text{ cm}^{2}$ (b) $18.7$ cm

Question 12

Paper cups are cones, open at the top, each with a radius of $3.5$ cm and a slant height of $9$ cm. Ignoring waste, how many cups can be made from $1 \text{ m}^{2}$ of paper?

ShowHide solution

One cup: $\pi \times 3.5 \times 9 = 98.96\ldots \text{ cm}^{2}\text{.}$

$1 \text{ m}^{2} = 100 \text{ cm} \times 100 \text{ cm} = 10\,000 \text{ cm}^{2}\text{.}$ $10\,000 \div 98.96\ldots = 101.05\ldots$

Only whole cups can be made, so round down.

$101$ cups

Key Points

The curved surface of a cone opens out into a sector of radius $l$ whose arc is $2\pi r$ long.
Curved surface area $= \pi r l$ ; total surface area of a solid cone $= \pi r l + \pi r^{2}$ .
The formulas use the slant height $l$ ; if the height is given, find $l$ from $l^{2} = r^{2} + h^{2}$ .
A hollow cone open at its base (a hat, a cup, a roof) has only its curved surface.
Add exact amounts and round only at the end.