AFCAT Numerical Ability · Surface Area and Volume

Cylinder

Volume and curved and total surface of a cylinder, a sheet rolled or a rectangle turned into one, hollow pipes, and how two cylinders compare.

All Surface Area and Volume notes

Why this matters

Nearly every question is πr²h or 2πrh with π as 22/7, and the arithmetic is kind when the radius is a multiple of 7 or 3.5. The marks go on a diameter read as a radius, and on which surfaces the story counts: a painted pillar is the curved surface only, a tank's wetted surface adds the floor, a closed drum has both ends.

Concept 1 of 4: Volume of a cylinder

A cylinder is a prism on a circle: base area

\pi r^2

times height. With

\pi = \tfrac{22}{7}

, a radius that is a multiple of 7 (or 3.5) cancels the 7 and leaves whole numbers.

Definition

Volume $\pi r^2 h$ . Halve a given diameter first.
Working back: $h = \dfrac{V}{\pi r^2}$ and $r^2 = \dfrac{V}{\pi h}$ .
Water: 1 cm³ holds 1 g (1 mL), and 1 m³ is 1,000 litres.
A rectangle turned about one of its sides makes a cylinder: that side is the height, the other side is the radius.

Cylinder of radius r and height h

V = \pi r^2 h

Worked example

A drum has diameter 42 cm and height 50 cm. How many litres does it hold? (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Surface Area and Volume · Cylinder

A figure is formed by revolving a rectangular sheet of dimensions 7 cm

\times

4 cm about its length. What is the volume of the figure thus formed?

Which side it turns about

A rectangle 10 cm × 3 cm turned about the 10 cm side gives radius 3 and height 10, volume

90\pi

. Turned about the 3 cm side it gives

300\pi

. The side it turns on is the height.

Halve the diameter

A cylinder 10 cm across and 6 cm tall holds

150\pi

cm³. Using 10 as the radius gives

600\pi

, four times too much.

Drill 4 more on volume of a cylinder

Concept 2 of 4: Curved and total surface, and the stories that use them

Unroll the curved surface of a cylinder and it is a rectangle: one side is the circumference

2\pi r

, the other is the height. So the curved surface is

2\pi rh

, and the two circular ends add

2\pi r^2

Definition

Curved surface $2\pi rh$ ; total surface $2\pi r(r + h)$ .
A sheet rolled into a cylinder: the side it is rolled along becomes the circumference $2\pi r$ , the other side the height.
Which surface: a painted pillar or a label is curved only; an open tank or a wetted surface is curved (up to the water) plus one base; a closed drum is total.
Rolling: one turn moves the cylinder one circumference along the ground and presses one curved surface of area.
A wire wound side by side to cover the curved surface: wire length × wire thickness = curved surface.
A flat cut tilted at angle $\theta$ to the circular cross-section is an ellipse of area $\dfrac{\pi r^2}{\cos\theta}$ .

Cylinder surfaces

\text{CSA} = 2\pi rh,\quad \text{TSA} = 2\pi r(r + h)

Worked example

A closed cylinder has radius 14 cm and height 6 cm. Find its curved and total surface. (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Surface Area and Volume · Cylinder

10 cylindrical pillars of a building have to be painted. The diameter of each pillar is 70 cm and the height is 4 m. What is the cost of painting at the rate of ₹5 per

\text{m}^2

A wetted surface has one base, not two

A tank of radius 7 m with water 5 m deep wets

220 + 154 = 374

m²: the wall up to the water and the floor. The lid and the dry wall are not wet.

Turn centimetres into metres before a rate per m²

A pillar 56 cm across has radius 0.28 m. Multiplying 28 by a height in metres gives an area a hundred times too big.

Drill 9 more on curved and total surface, and the stories that use them

Concept 3 of 4: Hollow cylinders and pipes

A pipe is a big cylinder with a smaller one taken out. Its material is the difference of the two volumes, and its surface has four parts: outside, inside and two rings at the ends.

Definition

Material $\pi h(R^2 - r^2)$ for outer radius $R$ and inner radius $r$ . The thickness is $R - r$ .
Total surface = outer $2\pi Rh$ + inner $2\pi rh$ + two end rings $2\pi(R^2 - r^2)$ .
Diameters given: halve both before squaring.

Hollow cylinder

V = \pi h (R^2 - r^2)

Worked example

A pipe 14 cm long has outer radius 9 cm and inner radius 7 cm. Find the volume of material and the total surface, in terms of

\pi

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Surface Area and Volume · Cylinder

The outer and inner diameters of a circular pipe are 6 cm and 4 cm, respectively. If its length is 10 cm, then what is the total surface area in

\text{cm}^2

R² − r² is not (R − r)²

With radii 9 and 7,

81 - 49 = 32

, but

(9 - 7)^2 = 4

. Square each radius first, then subtract.

A pipe's total surface has four parts

Outside, inside and the two end rings. Leaving out the inner wall or the rings is the usual way to land on a wrong option.

Drill 1 more on hollow cylinders and pipes

Concept 4 of 4: Comparing two cylinders, and changing their size

Volume goes as

r^2 h

and curved surface as

rh

, so a ratio question never needs

\pi

. Write both ratios, cancel, and the radius counts twice in volume but once in surface.

Definition

$\dfrac{V_1}{V_2} = \dfrac{r_1^2 h_1}{r_2^2 h_2}$ and $\dfrac{\text{CSA}_1}{\text{CSA}_2} = \dfrac{r_1 h_1}{r_2 h_2}$ .
Given the radius ratio and the surface ratio: find the height ratio from the surfaces first, then the volumes.
Equal volumes: $r_1^2 h_1 = r_2^2 h_2$ .
Volume divided by curved surface is $\dfrac{r}{2}$ , a quick way to the radius when both are given.
A wire (a long thin cylinder) of fixed volume: its length goes as $\dfrac{1}{r^2}$ .

Two cylinders

\dfrac{V_1}{V_2} = \dfrac{r_1^2 h_1}{r_2^2 h_2},\quad \dfrac{V}{\text{CSA}} = \dfrac{r}{2}

Worked example

Two cylinders have radii in the ratio

4 : 7

and heights in the ratio

7 : 4

. Compare their volumes and their curved surfaces.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Surface Area and Volume · Cylinder

The radii of two cylinders are in the ratio 2 : 3 and their curved surface areas are in the ratio 5 : 3. What is the ratio of their volumes?

The radius counts twice in volume, once in surface

Cylinders of equal height with radii

1 : 4

have curved surfaces

1 : 4

but volumes

1 : 16

A thinner wire grows by more than the cut

Cut the diameter by 20% and the cross-section becomes 0.64 of itself, so the length must grow to

\tfrac{1}{0.64} = 1.5625

times: up 56.25%, not 20%.

Drill 3 more on comparing two cylinders, and changing their size

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 (4)

Volume of a cylinder
Cylinder of radius r and height h
$V = \pi r^2 h$
Curved and total surface, and the stories that use them
Cylinder surfaces
$\text{CSA} = 2\pi rh,\quad \text{TSA} = 2\pi r(r + h)$
Hollow cylinders and pipes
Hollow cylinder
$V = \pi h (R^2 - r^2)$
Comparing two cylinders, and changing their size
Two cylinders
$\dfrac{V_1}{V_2} = \dfrac{r_1^2 h_1}{r_2^2 h_2},\quad \dfrac{V}{\text{CSA}} = \dfrac{r}{2}$

Watch out for (8)

Which side it turns about
→ Volume of a cylinder
Halve the diameter
→ Volume of a cylinder
A wetted surface has one base, not two
→ Curved and total surface, and the stories that use them
Turn centimetres into metres before a rate per m²
→ Curved and total surface, and the stories that use them
R² − r² is not (R − r)²
→ Hollow cylinders and pipes
A pipe's total surface has four parts
→ Hollow cylinders and pipes
The radius counts twice in volume, once in surface
→ Comparing two cylinders, and changing their size
A thinner wire grows by more than the cut
→ Comparing two cylinders, and changing their size

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