SSC CGL Maths · Mensuration 3D

Cylinders

A cylinder is a prism with a circle for its base: it holds πr²h, its curved side is 2πrh, and closing both ends adds two circles.

All Mensuration 3D notes

Why this matters

Twenty-two questions in the bank, seventeen EASY and five MODERATE, thirteen of them from the 2024 shifts. Most give one measure (the curved area, the circumference or the volume) and ask for another, so the skill is getting the radius out quickly. With π taken as 22/7, cancel the 7 before you multiply.

Concept 1 of 3: Volume and surface area of a cylinder

A cylinder is a prism whose base is a circle, so its volume is the circle's area times the height. Cut the curved side down a straight line and unroll it: you get a rectangle whose sides are the circumference and the height.

Definition

Volume $= \pi r^2 h$ .
Curved surface $= 2\pi rh$ : circumference times height.
Total surface of a closed cylinder $= 2\pi rh + 2\pi r^2 = 2\pi r(r + h)$ . A vessel open at the top has only one circle: $2\pi rh + \pi r^2$ .
Ratio of total to curved surface $= (r + h) : h$ , with no $\pi$ to work out.
With $\pi = \frac{22}{7}$ , cancel the 7 against a radius or height that is a multiple of 7 before you multiply. If a question says 3.14, use 3.14, because the options are worked with it.

Cylinder

V = \pi r^2 h \qquad \text{CSA} = 2\pi rh \qquad \text{TSA} = 2\pi r(r + h)

$r$ radius of the base
$h$ height

Worked example

A closed cylindrical drum has radius 7 cm and height 15 cm. Find its volume, its curved surface area and its total surface area. (Take

\pi = \frac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 3 · Q65Moderate

Example 1 · Mensuration 3D · Cylinders

A cylindrical storage container has a base radius of 3 cm. It is designed with sections whose heights form an arithmetic progression, starting from 2 cm for the bottom section and reaching 10 cm for the topmost section, covering a total of 5 sections. What is the total volume of the container? (Use

\pi \approx 3.14

)

Using the diameter as the radius

Many cylinder questions give the diameter. Halve it first: a flask 10 cm across has radius 5 cm. Putting 10 into

\pi r^2 h

gives four times the volume.

Two ends on an open vessel

A tank or a glass open at the top has one circular end, so its outer surface is

2\pi rh + \pi r^2

, not

2\pi r(r + h)

Drill 9 more on volume and surface area of a cylinder

Concept 2 of 3: Finding the radius from the curved surface or the circumference

Many questions hide the radius inside another measure. Take it out first: from the circumference by dividing by 2π, from the curved area by dividing by 2πh. After that it is the basic formula again.

Definition

From the circumference $C$ : $r = \frac{C}{2\pi}$ . With $\pi = \frac{22}{7}$ this is $r = \frac{7C}{44}$ .
From the curved area and the height: $r = \frac{\text{CSA}}{2\pi h}$ .
Shortcut: $\pi r^2 h = \frac{r}{2} \times 2\pi rh$ , so volume $=$ curved area $\times \frac{r}{2}$ . It is safe whenever the radius and the curved area belong to the same cylinder.
Rolling a sheet: the edge that goes round becomes the circumference and the other edge becomes the height. Rolled along the breadth means the breadth goes round.
Litres and level: $1000\ \text{cm}^3 = 1$ litre. Water drawn out of a cylinder lowers the level by (volume drawn out) $\div$ (base area).

Radius from the circumference, volume from the curved area

r = \frac{C}{2\pi} \qquad V = \frac{r}{2} \times \text{CSA}

Worked example

The curved surface area of a cylinder is

1320\ \text{cm}^2

and its height is 15 cm. Find its volume. (Take

\pi = \frac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 19 Sep 2024, Shift 2 · Q63Easy

Example 2 · Mensuration 3D · Cylinders

The base radius of a circular cylinder is 10.5 cm. If the area of its curved surface is 792

\text{cm}^2

, then what is the volume of the cylinder?

Rolling the sheet the wrong way

A sheet rolled along its breadth has the breadth as the circumference. Swap the two edges and you get a different cylinder with a different volume, so check which edge goes round.

Litres against cubic metres

1\ \text{m}^3

is 1000 litres, but 1 litre is

1000\ \text{cm}^3

. A level drop worked in centimetres with a volume in litres is off by 1000 unless you convert first.

Drill 7 more on finding the radius from the curved surface or the circumference

Concept 3 of 3: Hollow cylinders, pipes and drilled holes

A pipe is a big cylinder with a smaller one taken out of its middle. Both have the same length, so the metal is the ring at the end times the length.

Definition

Metal in a pipe of outer radius $R$ , inner radius $r$ and length $h$ : $\pi(R^2 - r^2)h = \pi(R + r)(R - r)h$ .
The water the pipe holds is the inner cylinder alone: $\pi r^2 h$ .
Share drilled out: with the same height, volumes compare as the squares of the radii. A hole of radius $r$ through a cylinder of radius $R$ removes $\frac{r^2}{R^2}$ of it; $k$ such holes remove $\frac{kr^2}{R^2}$ .
Read "the inner diameter equals the outer radius" slowly: it means $R = 2r$ .

Pipe and drilled share

V_{\text{metal}} = \pi(R^2 - r^2)h \qquad \text{share removed} = \frac{r^2}{R^2}

Worked example

A metal pipe is 20 cm long. Its outer radius is 4 cm and its inner radius is 3 cm. Find the volume of metal. (Take

\pi = \frac{22}{7}

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 3 · Q60Easy

Example 3 · Mensuration 3D · Cylinders

A solid cylinder has a radius of 5 cm and a height of 10 cm. A smaller cylindrical hole of radius 3 cm is drilled coaxially through its entire length. What is the approximate volume of the remaining solid?

Subtracting the radii before squaring

\pi(R - r)^2 h

is not the metal in a pipe. Square each radius first:

R^2 - r^2 = (R + r)(R - r)

. For radii 4 and 3 that is 7, not 1.

Drill 3 more on hollow cylinders, pipes and drilled holes

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)

Volume and surface area of a cylinder
Cylinder
$V = \pi r^2 h \qquad \text{CSA} = 2\pi rh \qquad \text{TSA} = 2\pi r(r + h)$
Finding the radius from the curved surface or the circumference
Radius from the circumference, volume from the curved area
$r = \frac{C}{2\pi} \qquad V = \frac{r}{2} \times \text{CSA}$
Hollow cylinders, pipes and drilled holes
Pipe and drilled share
$V_{\text{metal}} = \pi(R^2 - r^2)h \qquad \text{share removed} = \frac{r^2}{R^2}$

Watch out for (5)

Using the diameter as the radius
→ Volume and surface area of a cylinder
Two ends on an open vessel
→ Volume and surface area of a cylinder
Rolling the sheet the wrong way
→ Finding the radius from the curved surface or the circumference
Litres against cubic metres
→ Finding the radius from the curved surface or the circumference
Subtracting the radii before squaring
→ Hollow cylinders, pipes and drilled holes

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 22 questions