Volume and Surface Area of a Cone

Use the radius, height and slant of a cone to find its volume (one third of πr²h) and surface area (πrl plus the base), work back from a volume, and turn a sector into a cone.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1, IGCSE Additional Maths 0606, Edexcel Further Pure Maths 4PM1 · 28 real past-paper questions below

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Radius, Height and Slant

A cone has three lengths. The radius goes across the base. The height goes straight up the middle. The slant goes down the sloping side.

Formula

V = \frac{1}{3} π r^{2} h curved = π r l

$r$ is the radius, $h$ the height and $l$ the slant. The three make a right-angled triangle, so Pythagoras links them.

Worked example

1
$r = 5, h = 12$ — The question gives the radius and the height. We need the slant for the curved surface.
2
$l^{2} = 5^{2} + 1 2^{2}$ — The slant is the longest side of the triangle, so it is the one on its own. Add the squares of the other two.
3
$l^{2} = 25 + 144 = 169$ — Square each length first, then add.
4
$l = 169 = 13$ — Square root to undo the square. The slant is 13.

Exams also give the slant and ask for the height. Now the slant is the side you take away from.

Worked example

1
$l = 17, r = 8$ — This time we know the slant and the radius. We want the height.
2
$h^{2} + 8^{2} = 1 7^{2}$ — Same triangle: the two short sides squared add up to the slant squared.
3
$h^{2} = 289 - 64 = 225$ — Take the radius squared away from both sides: $1 7^{2} = 289$ and $8^{2} = 64$ .
4
$h = 225 = 15$ — Square root. The height is 15. Use only the positive root, because a length cannot be negative.

Common mistake

Swapping them. Volume uses the height; the curved surface uses the slant.

Volume: a Third of a Cylinder

The volume question: a cone holds one third of the cylinder with the same base and height.

Worked example

1
$V = \frac{1}{3} π r^{2} h$ — Start from the formula. It needs the height, not the slant.
2
$V = \frac{1}{3} π \times 5^{2} \times 12$ — Put in radius 5 and height 12. The slant 13 is not used here.
3
$= \frac{1}{3} π \times 25 \times 12$ — Square the radius first: $5^{2} = 25$ .
4
$= \frac{1}{3} π \times 300$ — Multiply the numbers: 25 × 12 = 300.
5
$= 100 π$ — A third of 300 is 100. If the question says “in terms of π”, stop here.
6
$\approx 314.2 cm^{3}$ — 100 × π = 314.159…, so 314.2 to 1 decimal place. Volume is in cubic units.

Working back: a cone has volume 400 cm³ and height 12 cm. Find the radius.

Worked example

1
$\frac{1}{3} π r^{2} \times 12 = 400$ — Put the volume and the height into the formula. Only $r$ is unknown.
2
$4 π r^{2} = 400$ — A third of 12 is 4, so the left side is $4 π r^{2}$ .
3
$r^{2} = \frac{400}{4 π} = 31.83 \dots$ — Divide both sides by $4 π$ to get $r^{2}$ on its own. Keep every digit on the calculator.
4
$r = 31.83 \dots \approx 5.64 cm$ — Square root to undo the square. Take the positive root: a radius cannot be negative.

Asked for the height instead? Same start, but now divide by $\frac{1}{3} π r^{2}$ to get $h$ on its own.

Common mistake

Using the slant 13 as the height. That gives about 340 cm³, not 314.

Common mistake

Using the diameter as the radius. If the question says the base is 10 cm across, the radius is 5 cm. Halve it first.

Curved and Total Surface Area

The surface question: the curved part is $π r l$ . A solid cone also has a flat circle base. Textbooks print the total like this:

Formula

total = π r l + π r^{2} = π r (l + r)

$π r l$ is the curved side, with $l$ the slant. $π r^{2}$ is the base circle, with $r$ the radius. The last form takes out $π r$ from both parts. All three give the same answer.

Worked example

1
$curved = π \times 5 \times 13$ — The curved part uses the slant 13 from the first card, not the height 12.
2
$= 65 π$ — 5 × 13 = 65.
3
$base = π \times 5^{2} = 25 π$ — The flat base is a circle, so its area is $π r^{2}$ .
4
$total = 65 π + 25 π = 90 π$ — A solid cone has both parts, so add them. Both have π, so just add 65 + 25.
5
$\approx 282.7 cm^{2}$ — 90 × π = 282.74…. Area is in square units.

Key idea

Check with the short form: $π \times 5 \times (13 + 5) = 90 π$ . Same answer.

Common mistake

Adding the base when the question says curved surface only, or leaving it off for a solid cone. Read which one is asked.

Sector to Cone (Extended/Higher)

The 6-mark question: a sector is rolled up into a cone. The sector's radius becomes the slant. Its arc wraps round the base circle.

A sector has radius 10 cm and angle $21 6^{\circ}$ . Find the volume of the cone.

Worked example

1
$arc = \frac{216}{360} \times 2 π \times 10$ — The arc is $\frac{216}{360}$ of the full circle of radius 10.
2
$= 12 π$ — 216 ÷ 360 = 0.6, and 0.6 × 20π = 12π.
3
$2 π r = 12 π$ — The arc goes once round the base, so it equals the base circumference $2 π r$ .
4
$r = 6$ — Divide both sides by 2π. The base radius is 6.
5
$l = 10$ — The sector's radius 10 becomes the slant of the cone.
6
$h^{2} = 1 0^{2} - 6^{2} = 64$ — The slant is the longest side, so take the radius squared away: 100 − 36 = 64.
7
$h = 8$ — Square root. The height is 8 (a length, so the positive root).
8
$V = \frac{1}{3} π \times 6^{2} \times 8$ — Volume uses the height 8, not the slant 10.
9
$= 96 π \approx 301.6 cm^{3}$ — 36 × 8 = 288, and a third of 288 is 96. Then 96 × π = 301.59….

Key idea

Show every line. Most of the marks are for the steps: the radius, the height, then the volume. If one number is wrong, the later steps done the right way can still score.

Cone to Sector Angle (Extended/Higher)

Exams also go the other way: you know the cone and must find the angle of the sector it opens into.

Formula

θ = \frac{r}{l} \times 36 0^{\circ}

$θ$ is the sector angle, $r$ the base radius of the cone and $l$ its slant. Why: the arc is the base circumference $2 π r$ , and the full circle would be $2 π l$ . So the sector is $\frac{r}{l}$ of the full circle.

A cone has height 24 cm and volume $392 π$ cm³. It is opened out into a sector. Find the sector angle.

Worked example

1
$\frac{1}{3} π r^{2} \times 24 = 392 π$ — First find the radius. Put the volume and the height into the volume formula.
2
$8 π r^{2} = 392 π$ — A third of 24 is 8.
3
$r^{2} = 49$ — Divide both sides by 8π: 392 ÷ 8 = 49.
4
$r = 7$ — Square root. The radius is 7 (a length, so the positive root).
5
$l^{2} = 7^{2} + 2 4^{2} = 625$ — Now the slant. It is the longest side, so add the squares: 49 + 576 = 625.
6
$l = 25$ — Square root. The slant is 25. This is the sector's radius.
7
$θ = \frac{7}{25} \times 360$ — The sector is 7 out of 25 parts of the full circle.
8
$θ = 100. 8^{\circ}$ — 7 ÷ 25 = 0.28, and 0.28 × 360 = 100.8.

Key idea

Check: the arc is $\frac{100.8}{360} \times 2 π \times 25 = 14 π$ , and the base circumference is $2 π \times 7 = 14 π$ . They match.

Common mistake

Using the height 24 instead of the slant 25. The sector's radius is always the slant.

Which Method? Read the Question

Check which of the three lengths you have.

You see	You do
Radius and height, find volume	use $\frac{1}{3} π r^{2} h$
Curved surface area	use $π r l$ with the slant
Total surface, solid cone	add the base $π r^{2}$
Only two lengths given	Pythagoras for the third
Volume given, find the radius	put in, divide, square root
Volume given, find the height	put in, divide by $\frac{1}{3} π r^{2}$
A sector made into a cone	arc length equals base circumference
A cone opened into a sector, find the angle	$\frac{r}{l} \times 36 0^{\circ}$
Top cut off	Frustum of a Cone

Key idea

For the arc of a sector, see Arc Length. For the right-angled triangle, see Pythagoras' Theorem.

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