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 solids surface area of a cone surface area of cone the surface area of a cone

Mark the correct alternative of the following.
A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to that of the cone is?

  1. $3:5$
  2. $2:5$
  3. $3:1$
  4. $1:3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

It is given that, the volumes of both cylinder and cone are the same.

So, let volume of the cylinder and cone be $V.$
It is also given that, their base radii are the same.
So, let radius of the cylinder $=$ Radius of the cone $=r$
Let the height of the cylinder and the cone be $h _1$ and $h _2$ respectively.
Volume of cone $=\dfrac{1}{3}\pi r^2 h _2$
Volume of cylinder $=\pi r^2 h _1$
We know both volumes are same.
$\therefore$  $\dfrac{1}{3}\pi r^2 h _2=\pi r^2 h _1$

$\Rightarrow$  $\dfrac{h _1}{h _2}=\dfrac{\pi r^2}{3\pi r^2}$

$\therefore$  $\dfrac{h _1}{h _2}=\dfrac{1}{3}$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Mark the correct alternative of the following.
If the base radius and the height of a right circular cone are increased by $20\%$, then the percentage increase in volume is approximately.

  1. $60$
  2. $68$
  3. $73$
  4. $78$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let original radius of base and height are $R$ and $H$ respectively.
Original height $=H$

New radius $=\dfrac{120}{100}R=\dfrac{6}{5}R$

And new height $=\dfrac{6}{5}H$

Original volume of cone $V _1=\dfrac{1}{3}\pi R^2 H$

New volume of cone $V _2=\dfrac{1}{3}\pi \left(\dfrac{6}{5}R\right)^2\times \dfrac{6}{5}H$

                                          $=\dfrac{216}{125}V _1$

Now, increased in volume $=\dfrac{216}{125}V _1-V _1=\dfrac{91}{125}V _1$

$\therefore$  Percentage increased in volume $=\left(\dfrac{91}{125}V _1\times\dfrac{1}{V _1}\right)\times 100\%$

                                                              $=72.81\%\approx 73\%$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Mark the correct alternative of the following.
The total surface area of a cone of radius $\dfrac{r}{2}$ and length $2l$, is?

  1. $2\pi r(l+r)$
  2. $\pi r\left(1+\dfrac{r}{4}\right)$
  3. $\pi r(l+r)$
  4. $2\pi rl$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $r$ and $l$ be base radius and slant height of cone.

Total surface area $=\pi r (l+r)$
Here, it is given that, 
The base radius is $\dfrac{r}{2}$ and that the slant height is $2l.$
Substituting these values in the above equation we have,
Total surface area $=\pi\left(\dfrac{r}{2}\right)\left(2l+\dfrac{r}{2}\right)$

                               $=\pi r\left(l+\dfrac{r}{4}\right)$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

If the radius of the base and the height of a right circular cone are respectively $21$ cm and $28$ cm, then the curved surface area of the cone is $\displaystyle \left(\pi\, =\, \frac{22}{7}\right)$

  1. $3696\, cm^{2}$
  2. $2310\, cm^{2}$
  3. $2550\, cm^{2}$
  4. $2410\, cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a cone, $l = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $ where h is the height. 
Hence, $l = \sqrt { { 28 }^{ 2 }+  {21}^{ 2 } } $

$ l = 35 $ cm 

Curved surface area of a cone $= \pi rl$  where r is the radius of the cone and $l$ is the slant height.
Hence, curved surface area of this cone $ = \dfrac {22}{7} \times 21 \times 35 = 2310  {cm}^{2} $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A conical tent with base-radius $7$ m and height $24$ m is made from $5$ m wide canvas. The length of the canvas used is $\displaystyle \left( \pi\, =\, \frac{22}{7}\right)$

  1. $100$ m
  2. $105$ m
  3. $110$ m
  4. $115$ m
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area of the canvas $ = $ Curved surface area of the conical tent
Since the canvas is rectangular in shape, its area is $=$ length $\times $ width
Curved surface area of a cone $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.
For a cone, $ l= \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $, where $l$ is the slant height.

Hence, $ l = \sqrt { { 24 }^{ 2 }+  {7}^{ 2 } } $ 

$\Rightarrow  l = \sqrt {625} $

$ \Rightarrow l = 25 $ cm

Hence, length $ \times 5 =\displaystyle \frac { 22 }{ 7 } \times 7\times 25 $

$ \therefore $ length $= 110 $ m

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

If the radius and slant height of a cone are in the ratio $4 : 7$ and its curved surface area is $792 cm^{2}$, then its radius is $\displaystyle \left(\pi\, =\, \frac{22}{7}\right)$.

  1. $10$ cm
  2. $8$ cm
  3. $12$ cm
  4. $9$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the radius of the cone be $ 4a$ and slant height $ = 7a$ 

Given, curved surface area $=792$ $cm^2$
Curved surface area of a cone $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.

Hence, curved surface area of this cone $ = \displaystyle \frac {22}{7} \times 4a \times 7a = 792 $
$ \therefore  a = 3 $ cm 

Hence, radius $ = 4a  = 12 $ cm 

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

If the curved surface area of a right circular cone is $12,320 cm^{2}$ and its base radius is $56$ cm, then its height is $\displaystyle \left(\pi\, =\, \frac{22}{7}\right)$

  1. $42$ cm
  2. $36$ cm
  3. $48$ cm
  4. $50$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Curved surface area of a cone $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.
Hence, curved surface area of this cone, $ = \dfrac {22}{7} \times 56 \times l = 12320 $ 

$ \therefore l = 70  cm $

For a cone, $ l = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $, where $h$ is the height. 

Hence, $ 70 = \sqrt { { h }^{ 2 }+  {56}^{ 2 } } $ 

$\Rightarrow  4900 = { h }^{ 2 } + 3136 $

$\Rightarrow  { h }^{ 2 } = 1764 $

$ \Rightarrow  h = 42 $ cm

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A joker's cap is in the form of a right circular cone of base radius $7$ cm and height $24$ cm. Find the area of the sheet required to make $100$ such caps.

  1. $55000{cm}^{2}$
  2. $48724{cm}^{2}$
  3. $30000{cm}^{2}$
  4. $11256{cm}^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of sheet required to make a cap is the Curved surface area of a cone which is $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.

For a cone, $l = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $, where $h$ is the height. 

Hence, $ l = \sqrt { { 24 }^{ 2 }+  {7}^{ 2 } } $ 

$ \therefore l = 25 $ cm

Hence, area of sheet required to make one cap $

= \dfrac {22}{7} \times 7 \times 25 = 550 $ sq.cm  
Area of sheet required to make $100$ caps $ 100\times 550=55000$ sq.cm

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The area of the base of a cone is $616$ sq.cm. Its height is $48cm$. What is its total surface area ?

  1. $2816{cm}^{2}$
  2. $2861{cm}^{2}$
  3. $2618{cm}^{2}$
  4. $2681{cm}^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The base of a cone is circular. 

Area of a circle  $ = \pi { r }^{ 2 } = 616 $
$\Rightarrow r^2 = \cfrac {616 \times 7}{22} = 196$
$ => r = 14  cm $

Total surface area of a cone $ = \pi r (r + l) $ 
where,
$r$ is the radius of the cone and 
$l$ is the slant height.

For a cone, $l = \sqrt {r^2+h^2}$ 

where, $h$ is the perpendicular height. 


Hence, $l = \sqrt { { 48}^{ 2 }+  {14}^{ 2 } } \sqrt {2500} $

$ l = 50  cm $

Hence, TSA of this cone, $ = \cfrac {22}{7} \times 14 \times (14 + 50) = 2816  {cm}^{2} $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The radius of a cone is $3\ cm$ and vertical height is $4\ cm$. Find the area of the curved surface.

  1. $62.85\ {cm}^{2}$
  2. $66.05\ {cm}^{2}$
  3. $52.25\ {cm}^{2}$
  4. $47.14\ {cm}^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Curved surface area of a cone $= \pi rl$

For a cone, $l = \sqrt { {h }^{ 2 }+  {r}^{ 2 } } $

Hence, $l = \sqrt { { 3 }^{2 }+  {4}^{ 2 } } $

$ l = 5  cm $
So, CSA of this cone, $ = \cfrac {22}{7} \times 3 \times 5 = 47.14  {cm}^{2} $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The radius and height of a cone are in the ratio $4:3$. The area of the base is $154\space cm^2$. Find the area of the curved surface.

  1. $192.5\space cm^2$
  2. $195\space cm^2$
  3. $190.5\space cm^2$
  4. $185.5\space cm^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let radius be $4x$ and height be $3x$.


$\therefore $ Area of base $=$ $\pi { r }^{ 2 }=154\Rightarrow \dfrac { 22 }{ 7 } \times { \left( 4x \right)  }^{ 2 }=154$


$\Rightarrow \quad 16{ x }^{ 2 }=\dfrac { 154\times 7 }{ 22 } \Rightarrow { x }^{ 2 }=\dfrac { 49 }{ 16 } \Rightarrow x=\dfrac { 7 }{ 4 } $

$\therefore $ Radius $=$ $4\times \dfrac { 7 }{ 4 } =7cm,\quad height=3\times \dfrac { 7 }{ 4 } =\dfrac { 21 }{ 4 } =5.25cm$

Now, CSA of cone $=$ $\pi rl\quad and\quad l=\sqrt { { r }^{ 2 }+{ h }^{ 2 } } =\sqrt { 49+{ \left( 5.25 \right)  }^{ 2 } } $

$\therefore $ CSA $=$ $\dfrac { 22 }{ 7 } \times 7\times 8.75=192.5{ cm }^{ 2 }$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of of a right cone is $\displaystyle 286m^{2}$ and its slant height is 13 m then area of the base is

  1. $\displaystyle 286m^{2}$
  2. $\displaystyle 308m^{2}$
  3. $\displaystyle 154m^{2}$
  4. $\displaystyle 187m^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Curved surface area of a cone $= \pi rl$  where r is the radius of
the cone and l is the slant height.
Hence, CSA of this cone, $ = \frac {22}{7} \times r \times 13 = 286 $
$ =>r = 7  m $

Area of base $ = \pi {r}^{2} = \frac {22}{7} \times 7 \times 7 = 154  m^2 $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The total surface area of cone if its slant height is 9 m, and the radius of its base is 12 m is

  1. 525 $\displaystyle cm^{2}$
  2. 792 $\displaystyle cm^{2}$
  3. 684 $\displaystyle cm^{2}$
  4. 412 $\displaystyle cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We know that the total surface area S of a right circular cylinder of radius r and slant height l is given by 
$\displaystyle S=\pi r^{2}+\pi rl=\pi r\left ( r+l \right )$
Here, $\displaystyle r=12m\, and\, l=9m$
$\displaystyle \therefore S=\left { \frac{22}{7}\times 12\times \left ( 12+9 \right ) \right }^{2}=792m^{2}$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of a cone of slant height l and radius r is given by

  1. $ \displaystyle \frac{1}{3}\pi / r^{2} $
  2. $ \displaystyle \pi rl $
  3. $ \displaystyle \pi rl^{2} $
  4. $ \displaystyle \frac{1}{3}\pi rl $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A cone is a three-dimensional geometric shape consisting of all line segments joining a single point to every point of a two-dimensional figure.

Slant height of cone (l)=$\sqrt{r^{2}+h^{2}}$Them covered surface area of cone =$\pi rl$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A cylindrical rod of lenght h is meted and cast into a cone of base radius twice that of the cylinder What is the height of the cone?

  1. $\displaystyle \frac{3h}{4}$
  2. $\displaystyle \frac{4h}{4}$

  3. 2h
  4. $\displaystyle \frac{h}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the radius of  cylindrical rod is r and  cylindrical rod height is h given

Then volume of  cylindrical rod=$\pi r^{2}h$
And volume of cone of base twice the radius of   cylindrical rod=$\frac{1}{3}\pi (2r)^{2}H=\frac{4}{3}\pi r^{2}H$
The cylindrical rod melted and make cone 
$\therefore \frac{4}{3}\pi r^{2}H=\pi r^{2}h\Rightarrow H=\frac{3h}{4}$