CDS Mathematics · Mensuration 3D

Cylinders

Volume, curved surface and total surface of a cylinder, a sheet rolled into a cylinder, and hollow pipes.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

Mostly MODERATE and very regular: the paper fixes a ratio of radius to height, or of curved to total surface, and one number. The ratio is the real information; turn it into a relation between r and h first.

Concept 1 of 3: Volume and surface of a cylinder

Unroll the curved surface and it is a rectangle: the circumference by the height. Add the two circular ends for the total surface. Ratio conditions then collapse to one line: curved : total

= h : (r + h)

Definition

Volume $\pi r^2 h$ ; curved surface $2\pi rh$ ; total surface $2\pi r(r + h)$ .
Curved : total $= h : (r + h)$ . Curved $=$ half the total means $r = h$ .
Radius : height $= p : q$ : write $r = pk$ , $h = qk$ and solve for $k$ .
Given $r + h$ and $rh$ (from the curved surface): $r$ and $h$ are the roots of a quadratic.

Cylinder

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

Worked example

The radius and height of a cylinder are in the ratio

1 : 2

and its volume is

2156

^3

. Find its total surface area.

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

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q43Moderate

Example 1 · Mensuration 3D · Cylinders

The ratio of the curved surface area to the total surface area of a right circular cylinder is 1 : 2. If the total surface area is

616\ \text{cm}^2

, what is the volume of the cylinder ?

Diameter or radius — again

Stems give the diameter of a well, a box or a pipe. Halve it before squaring; the option built on the full diameter is four times too big.

Drill 9 more on volume and surface of a cylinder

Concept 2 of 3: Rolling a sheet into a cylinder

Roll a rectangle and one side becomes the circumference, the other the height. The two ways of rolling give different volumes: the longer side as the circumference gives the fatter, larger cylinder.

Definition

Sheet $a\times b$ rolled so that $a$ is the circumference: $r = \dfrac{a}{2\pi}$ , $h = b$ , $V = \dfrac{a^2 b}{4\pi}$ .
The two rollings have volumes in the ratio $\dfrac{a^2 b}{a b^2} = \dfrac ab$ : longer side round gives more.
The curved surface is the sheet's area either way.

Sheet rolled along side a

V = \frac{a^2 b}{4\pi}

Worked example

66

cm by

22

cm sheet is rolled so that the

66

cm side forms the circumference. Find the volume.

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

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q69Moderate

Example 2 · Mensuration 3D · Cylinders

A rectangular paper is 44 cm long and 22 cm wide. Let

x

be the volume of the largest cylinder formed by rolling the paper along its length and

y

be the volume of the largest cylinder formed by rolling the paper along its width. What is the ratio of

x

y

? (Take

\pi = \frac{22}{7}

)

Drill 3 more on rolling a sheet into a cylinder

Concept 3 of 3: Hollow cylinders and pipes

A pipe is a big cylinder minus a small one. Its metal is

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

, which factorises as

\pi(R + r)(R - r)h

— so a question that gives the thickness

R - r

and the metal volume hands you

R + r

Definition

Metal $= \pi(R^2 - r^2)h$ .
Outer curved minus inner curved $= 2\pi(R - r)h$ .
All surfaces of a hollow cylinder: outer curved $+$ inner curved $+$ two ring-shaped ends $2\pi(R^2 - r^2)$ .

Hollow cylinder

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

Worked example

A pipe

21

cm long has outer radius

5

cm and inner radius

4

cm. Find the volume of metal.

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

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q64Moderate

Example 3 · Mensuration 3D · Cylinders

The difference between the outside and the inside surface area of a cylindrical pipe 14 cm long is 44 cm

^2

. The pipe is made of 99 cm

^3

of metal. If

R

is the outer radius and

r

is the inner radius of the pipe, then what is

(R + r)

equal to? (Take

\pi = \frac{22}{7}

)

Drill 1 more on hollow cylinders and pipes

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 of a cylinder
Cylinder
$V = \pi r^2 h, \quad \text{CSA} = 2\pi r h, \quad \text{TSA} = 2\pi r(r + h)$
Rolling a sheet into a cylinder
Sheet rolled along side a
$V = \frac{a^2 b}{4\pi}$
Hollow cylinders and pipes
Hollow cylinder
$V = \pi(R^2 - r^2)h = \pi(R + r)(R - r)h$

Watch out for (1)

Diameter or radius — again
→ Volume and surface of a cylinder

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 16 questions