SSC CGL Maths · Mensuration 3D

Cones

A cone holds one third of the cylinder on the same base and height, its curved surface is π times the radius times the slant height, and radius, height and slant height form one right triangle.

All Mensuration 3D notes

Why this matters

Nineteen questions in the bank, seventeen EASY and two MODERATE. Nearly every one turns on a single right triangle: radius, height and slant height. Find whichever of the three is missing, then use the height for the volume and the slant height for the surface.

Concept 1 of 3: Radius, height and slant height of a cone

Slice a cone straight down through its tip and you see a right triangle. The height stands up the middle, the radius runs along the base, and the slant height is the sloping side, the longest of the three.

Definition

$l^2 = r^2 + h^2$ , so $l = \sqrt{r^2 + h^2}$ and $h = \sqrt{l^2 - r^2}$ .
Know the common triples and their multiples: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25; 20, 21, 29. A multiple such as 6, 8, 10 or 9, 12, 15 works the same way.
Turning a right triangle about one of its shorter sides makes a cone: that side is the height, the other short side is the radius, and the longest side is the slant height.
Two special cones: if $h = r$ , then $l = r\sqrt{2}$ ; if $l = 2r$ , then $h = r\sqrt{3}$ .

Slant height

l = \sqrt{r^2 + h^2}

$r$ radius of the base
$h$ vertical height
$l$ slant height

Worked example

A cone has radius 9 cm and height 12 cm. Find its slant height.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 3 · Q69Easy

Example 1 · Mensuration 3D · Cones

A right circular cone has a radius of 7 cm and height of 24 cm. What is the slant height?

Slant height used as the height

The slant height is the sloping edge; the height runs from the tip straight down to the centre of the base. With

r = 6

and

l = 10

the height is 8, not 10.

Turning about the wrong side

A 6, 8, 10 triangle turned about the 8 cm side gives a cone of height 8 cm and radius 6 cm. The side it turns about is always the height.

Drill 5 more on radius, height and slant height of a cone

Concept 2 of 3: Volume of a cone

A cone holds one third of the cylinder with the same base and height, just as a pyramid holds one third of its prism.

Definition

$V = \frac{1}{3}\pi r^2 h$ .
Working back: $r^2 = \frac{3V}{\pi h}$ and $h = \frac{3V}{\pi r^2}$ .
When the slant height is given, find the height first: $h = \sqrt{l^2 - r^2}$ .
Fast arithmetic: with $\pi = \frac{22}{7}$ , $\frac{1}{3}\pi r^2 h = \frac{22 r^2 h}{21}$ . Cancel the 7 and the 3 against $r^2 h$ before you multiply.

Volume of a cone

V = \frac{1}{3}\pi r^2 h

Worked example

A cone has radius 6 cm and height 14 cm. Find its volume. (Take

\pi = \frac{22}{7}

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 13 Sep 2024, Shift 1 · Q75Easy

Example 2 · Mensuration 3D · Cones

What is the height (in m) of a conical water tank having radius of 70 m and volume of

46,200\ \text{m}^3

? (Take

\pi = \frac{22}{7}

)

Leaving out the one third

A cone holds one third of the matching cylinder.

\pi r^2 h

alone gives three times the cone's volume, and that value is often among the options.

Slant height in the volume formula

The volume needs the vertical height. When a question gives the slant height, find

h = \sqrt{l^2 - r^2}

before you use

\frac{1}{3}\pi r^2 h

Drill 3 more on volume of a cone

Concept 3 of 3: Curved and total surface area of a cone

Cut a paper cone down its side and flatten it: you get a sector of a circle whose radius is the slant height. Its area works out to π times the base radius times the slant height.

Definition

Curved surface $= \pi rl$ . It needs the slant height, not the height.
Total surface of a solid cone $= \pi rl + \pi r^2 = \pi r(l + r)$ .
A tent or a paper cone has no base: use the curved surface alone.
If the circumference $C$ of the base is given, $r = \frac{C}{2\pi}$ .
"Curved area is $k$ times the base area" means $\pi rl = k\pi r^2$ , so $l = kr$ .

Surface of a cone

\text{CSA} = \pi rl \qquad \text{TSA} = \pi r(l + r)

Worked example

A solid cone has radius 5 cm and height 12 cm. Find its curved surface area and its total surface area in terms of

\pi

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 26 Sep 2024, Shift 1 · Q70Easy

Example 3 · Mensuration 3D · Cones

What is the total surface area of a cone whose curved surface area is 550

\text{cm}^2

and radius is 7 cm?

\left(\text{Take } \pi = \frac{22}{7}\right)

Adding a base to a tent

A tent or a cap has no base. Adding

\pi r^2

asks for canvas that is not there; use

\pi rl

alone.

Height in the curved surface

\pi rh

is not a cone formula. The curved surface uses the slant height:

\pi rl

Drill 8 more on curved and total surface area of a cone

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 and slant height of a cone
Slant height
$l = \sqrt{r^2 + h^2}$
Volume of a cone
Volume of a cone
$V = \frac{1}{3}\pi r^2 h$
Curved and total surface area of a cone
Surface of a cone
$\text{CSA} = \pi rl \qquad \text{TSA} = \pi r(l + r)$

Watch out for (6)

Slant height used as the height
→ Radius, height and slant height of a cone
Turning about the wrong side
→ Radius, height and slant height of a cone
Leaving out the one third
→ Volume of a cone
Slant height in the volume formula
→ Volume of a cone
Adding a base to a tent
→ Curved and total surface area of a cone
Height in the curved surface
→ Curved and total surface area of a cone

Test yourself on Mensuration 3D

20 past SSC CGL questions from this chapter, timed at 12 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 19 questions