CDS Mathematics · Mensuration 3D

Cones

Volume and surface of a cone, the right triangle formed by its radius, height and slant height, and cones made by rolling a sector or turning a triangle.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

Cone questions are one right triangle wearing different clothes. Draw the axial section — radius, height, slant height, and the half-angle at the vertex — and nearly every question becomes Pythagoras or a trigonometric ratio.

Concept 1 of 3: Radius, height, slant height — one right triangle

Cut a cone down its axis and you see a right triangle with legs

r

and

h

and hypotenuse

l

. The volume uses

h

; the curved surface uses

l

. Mixing them up is the commonest error on this page.

Definition

$l^2 = r^2 + h^2$ .
Volume $\dfrac13\pi r^2 h$ (vertical height).
Curved surface $\pi r l$ (slant height); total $\pi r(l + r)$ .
Total : curved $= (l + r) : l$ .
A canvas tent is the curved surface only; cloth of width $w$ needs length $\dfrac{\pi r l}{w}$ .

Cone

V = \tfrac13\pi r^2 h, \quad \text{CSA} = \pi r l, \quad l^2 = r^2 + h^2

l² = r² + h², sin α = r ÷ l. Volume ⅓πr²h uses h; curved surface πrl uses l.

Worked example

A cone has radius

7

cm and slant height

25

cm. Find its volume.

(\pi = \tfrac{22}{7})

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q66Easy

Example 1 · Mensuration 3D · Cones

The radius and slant height of a right circular cone are 5 cm and 13 cm respectively. What is the volume of the cone ?

Volume uses h, surface uses l

Putting the slant height into

\dfrac13\pi r^2 h

is the planted error — with

r = 9

and

l = 15

it gives

405\pi

instead of

324\pi

Drill 10 more on radius, height, slant height — one right triangle

Concept 2 of 3: The vertical angle and the axial section

The vertical angle is split in two by the axis. The half-angle sits at the apex of the right triangle, opposite the radius, so

\sin = \dfrac rl

and

\tan = \dfrac rh

Definition

For semi-vertical angle $\alpha$ (half the vertical angle):

$r = l\sin\alpha$ , $h = l\cos\alpha$ , $r = h\tan\alpha$ .
Vertical angle $90^\circ$ : $r = h$ . Vertical angle $60^\circ$ : $r = \dfrac l2$ . Vertical angle $120^\circ$ : $h = \dfrac l2$ , $r = \dfrac{\sqrt3}{2}l$ .
An equilateral axial section of side $a$ : $r = \dfrac a2$ , $h = \dfrac{\sqrt3}{2}a$ .

Semi-vertical angle α

r = l\sin\alpha, \qquad h = l\cos\alpha

Worked example

A cone has a vertical angle of

90^\circ

and slant height

6\sqrt2

cm. Find its volume.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (I) 2025 — Elementary Mathematics · Q70Moderate

Example 2 · Mensuration 3D · Cones

The angle at the vertex of a conical body is

120^\circ

If the sum of slant height, height and radius is

(9 + 3\sqrt{3})

cm, then what is the volume of the cone ?

Halve the vertical angle first

The trigonometry uses the half-angle at the apex. Taking

\sin 120^\circ

instead of

\sin 60^\circ

happens to give the same value, which hides the slip until a

90^\circ

60^\circ

cone exposes it.

Drill 4 more on the vertical angle and the axial section

Concept 3 of 3: Cones from a sector or a turning triangle

Roll a sector of a circle and its radius becomes the slant height while its arc becomes the base circumference. Spin a right triangle about one leg and that leg becomes the height, the other the radius.

Definition

Sector of radius $R$ and angle $\theta$ rolled up: $l = R$ , $2\pi r = \dfrac{\theta}{360^\circ}2\pi R$ , so $r = \dfrac{\theta}{360^\circ}R$ .
A semicircle makes a cone with $r = \dfrac R2$ : the half-angle is $30^\circ$ .
A right triangle turned about a leg: that leg is the height, the other leg the radius, the hypotenuse the slant height.

Sector rolled into a cone

l = R, \qquad r = \frac{\theta}{360^\circ}\,R

Worked example

A sector of radius

12

cm and angle

120^\circ

is rolled into a cone. Find the cone's radius and height.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — Elementary Mathematics · Q74Moderate

Example 3 · Mensuration 3D · Cones

ABC

is a triangle right-angled at

B

with

AB = 8

cm and

BC = 6

cm. It is made to revolve about its side

BC

. What is the approximate total surface area of the cone so formed ? (take

\pi = \frac{22}{7}

)

Drill 1 more on cones from a sector or a turning triangle

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 (3)

Radius, height, slant height — one right triangle
Cone
$V = \tfrac13\pi r^2 h, \quad \text{CSA} = \pi r l, \quad l^2 = r^2 + h^2$
The vertical angle and the axial section
Semi-vertical angle α
$r = l\sin\alpha, \qquad h = l\cos\alpha$
Cones from a sector or a turning triangle
Sector rolled into a cone
$l = R, \qquad r = \frac{\theta}{360^\circ}\,R$

Watch out for (2)

Volume uses h, surface uses l
→ Radius, height, slant height — one right triangle
Halve the vertical angle first
→ The vertical angle and the axial section

Test yourself on Mensuration 3D

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Mensuration 3D chapter test Drill all 18 questions