Mathematics · Quantitative Aptitude

Mensuration of Solids

194 Questions

Mensuration of solids focuses on calculating the volume and surface area of three dimensional shapes like cylinders and cones. These geometry problems require applying standard mathematical formulas. They frequently appear in state public service and engineering exams.

Cylinder volume and areaCone slant heightSurface area ratiosHollow pipe calculationsDimensional optimization

Mensuration of Solids Questions

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

The ratio between the radius of the base and the height of a cylinder is $2:3$. If its volume is $12936$ cu. cm, the total  surface area of the cylinder is :

  1. $2587.2 c{m^2}$
  2. $3080 c{m^2}$
  3. $25872 c{m^2}$
  4. $38808 c{m^2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
We have $\dfrac{r}{h}=\dfrac{2}{3}\Rightarrow\,h=\dfrac{3r}{2}$

Volume of a cylinder$=\pi{r}^{2}h$

$\Rightarrow\,12936=\dfrac{22}{7}\times{r}^{2}\times \dfrac{3r}{2}$

$\Rightarrow\,12936=\dfrac{11\times 3}{7}{r}^{3}$

$\Rightarrow\,{r}^{3}=\dfrac{12936\times 7}{33}=2744$

$\Rightarrow\,r=\sqrt[3]{2744}=14\ cm$

We have $h=\dfrac{3r}{2}=\dfrac{3\times 14}{2}=21\ cm$

Total Surface area$=2\pi\,r\left(r+h\right)=2\times\dfrac{22}{7}\times 14\left(14+21\right)=2\times\dfrac{22}{7}\times 14\times 35=140\times 22=3080\ sq.cm$
Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

A cylindrical pipe is made from a metal sheet of length 88 cm and breadth 20 cm. What is the volume of this pipe?

  1. $2800$ ${ cm }^{ 3 }$
  2. $12320$ ${ cm }^{ 3 }$
  3. $13202$ ${ cm }^{ 3 }$
  4. $13220$ ${ cm }^{ 3 }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We have,

Length $l=88\,cm.$

Breadth$b=20\,cm.$

Volume $=?$

Volume of cylindrical pipe $=\pi {{r}^{2}}h$

We know that,

$ circumfrance=Breadth=2\pi r $

$ 2\pi r=20 $

$ \pi r=10 $

$ r=\dfrac{10}{\pi } $

$ r=\dfrac{10}{\dfrac{22}{7}} $

$ r=\dfrac{70}{22} $

$ r=\dfrac{35}{11}\,\,cm. $

Then,

Volume of cylindrical pipe $V=\pi {{r}^{2}}h$

$ V=\dfrac{22}{7}\times \dfrac{35}{11}\times \dfrac{35}{11}\times 88 $

$ V=2\times 5\times 35\times 8 $

$ V=80\times 35 $

$ V=2800\,c{{m}^{3}} $

Hence, this is the answer.

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

If sum of radius and height of a cylinder is 6, then its maximum volume is 

  1. $32\pi$
  2. $16\pi$
  3. $8\pi$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume V = pi * r^2 * h. Given r + h = 6, then h = 6 - r. V(r) = pi * r^2 * (6 - r) = pi * (6r^2 - r^3). To maximize, V'(r) = pi * (12r - 3r^2) = 0. So 3r(4 - r) = 0, r = 4. Then h = 2. Max Volume = pi * 4^2 * 2 = 32 * pi.

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
In a cylinder, if radius is doubled and height is halved, curved surface area will be?

  1. Halved

  2. Doubled

  3. Same

  4. Four times
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the radius of cylinder be $r$ and height be $h.$

So, the original curved surface area $=2\pi rh$
When, radius is doubled and height is halved,
New curved surface area $=2\pi \times 2r\times \dfrac{h}{2}$

                                           $=2\pi r h$
$\therefore$  New curved surface area $=$ Original surface area.
$\therefore$  There is no change in the curved surface area of the cylinder

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
If the height of a cylinder is doubled and radius remains the same, then volume will be?

  1. Doubled

  2. Halved

  3. Same

  4. Four times
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $V _1$ be the volume of the cylinder with radis $r _1$ and height $h _1,$ then

$\Rightarrow$  $V _1=\pi r _1^2 h _1$            ---- ( 1 )
Now, let $V _2$ be the volume after changing the dimension, then
$\Rightarrow$  $r _2=r _1,$  $h _2=2h _1$
So,
$\Rightarrow$  $V _2=\pi r _2^2h _2$

$\Rightarrow$  $V _2=\pi\times{r _1}^2\times 2h _1$

$\Rightarrow$  $V _2=2\times \pi r _1^2 h _1$
From ( 1 ),

$\Rightarrow$  $V _2=2V _1$

$\therefore$  If the height of a cylinder is doubled and radius remains the same, then volume will be $Doubled.$

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
In a cylinder, if radius is halved and height is doubled, the volume will be?

  1. Same

  2. Doubled

  3. Halved

  4. Four times
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $V _1$ be the volume of the cylinder with radis $r _1$ and height $h _1,$ then

$\Rightarrow$  $V _1=\pi r _1^2 h _1$            ---- ( 1 )
Now, let $V _2$ be the volume after changing the dimension, then
$\Rightarrow$  $r _2=\dfrac{1}{2}r _1,$  $h _2=2h _1$
So,
$\Rightarrow$  $V _2=\pi r _2^2h _2$

$\Rightarrow$  $V _2=\pi\times\left(\dfrac{r _1}{2}\right)^2\times 2h _1$

$\Rightarrow$  $V _2=\dfrac{1}{2}\times \pi r _1^2 h _1$
From ( 1 ),

$\Rightarrow$  $V _2=\dfrac{1}{2}V _1$

$\therefore$  In a cylinder, if radius is halved and height is doubled, the volume will be $Halved$

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
If the radius of a cylinder is doubled and the height remains same, the volume will be?

  1. Doubled

  2. Halved

  3. Same

  4. Four times
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $V _1$ be the volume of the cylinder with radius $r _1$ and height $h _1,$ then

$V _1=\pi r _1^2 h _1$           ---- ( 1 )
Now, let $V _2$ be the volume after changing the dimensions, then
$r _2=2r _1,\,h _2=h _1$
So,
$\Rightarrow$  $V _2=\pi r _2^2 h _2$

$\Rightarrow$  $V _2=\pi\times(2r _1)^2\times h _1$

$\Rightarrow$  $V _2=4\times \pi r _1^2 h _1$
From ( 1 ),
$\therefore$  $V _2=4V _1$
Hence, If the radius of a cylinder is doubled and the height remains same, the volume will be Four times.

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
The volume of a cylinder of radius r is $1/4$ of the volume of a rectangular box with a square base of side length x. If the cylinder and the box have equal heights, what is r in terms of x?

  1. $\dfrac{x^2}{2\pi}$
  2. $\dfrac{x}{2\sqrt{\pi}}$
  3. $\dfrac{\sqrt{2x}}{\pi}$
  4. $\dfrac{\pi}{2\sqrt{x}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the height of the cylinder be $h.$

Volume of the cylinder $=\pi r^2 h$
Height of the rectangular box $=h$
Since, base is square with side $x.$
Volume of the box $=x\times x\times h=x^2 h$
According to question,
$\Rightarrow$  $\pi r^2  h=\dfrac{1}{4} x^2 h$

$\Rightarrow$  $r^2=\dfrac{1}{4\pi}x^2$
Taking square root on both sides,
$\Rightarrow$  $r=\dfrac{x}{2\sqrt{\pi}}$

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
Two circular cylinders of equal volume have their heights in the ratio $1:2$. Ratio of their radii is?

  1. $1:\sqrt{2}$
  2. $\sqrt{2}:1$
  3. $1:2$
  4. $1:4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\dfrac{h _1}{h _2}=\dfrac{1}{2}$        [ Given ]
Let $V _1$ and $V _2$ are volume of cylinders.

$\therefore$  $V _1=V _2$          [ Given ]

$\therefore$  $\dfrac{V _1}{V _2}=1$

$\Rightarrow$  $\dfrac{\pi r _1^2h _1}{\pi r _2^2 h _2}=1$

$\Rightarrow$  $\left(\dfrac{r _1}{r _2}\right)^2\left(\dfrac{h _1}{h _2}\right)=1$

But it is given that,
$\dfrac{h _1}{h _2}=\dfrac{1}{2}$

$\therefore$  $\left(\dfrac{r _1}{r _2}\right)^2\times\dfrac{1}{2}=1$

$\Rightarrow$  $\left(\dfrac{r _1}{r _2}\right)^2=2$

$\Rightarrow$  $\left(\dfrac{r _1}{r _2}\right)^2=\dfrac{2}{1}$

$\Rightarrow$  $\dfrac{r _1}{r _2}=\dfrac{\sqrt{2}}{1}$

$\therefore$  The ratio of the radii of the two cylinders is $\sqrt{2}:1$
Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
The height h of a cylinder equal the circumference of the cylinder. In terms of h, what is the volume of the cylinder?

  1. $\dfrac{h^3}{4\pi}$
  2. $\dfrac{h^2}{2\pi}$
  3. $\dfrac{h^3}{2}$
  4. $\pi h^3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $h$ be the height of cylinder with radius $r.$

It is given that,
$2\pi r =h$
$\Rightarrow$  $r=\dfrac{h}{2\pi}$
Therefore, the volume of the cylinder is
$V=\pi r^2 h$

$\Rightarrow$  $V=\pi\left(\dfrac{h}{2\pi}\right)^2h$

$\Rightarrow$  $V=\dfrac{h^3}{4\pi}$