Mathematics · Quantitative Aptitude

Mensuration of Solids

209 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

If the base radius and slant height of a right circular cone are $10 \,cm$ and $3.5 \,cm$ respectively, then its total surface area is

  1. $424.159 cm^2$
  2. $434.159 cm^2$
  3. $414.159 cm^2$

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

Given the radius of the base of a right circular cone $(r)=10$ cm, and its slant height $(l)=3.5$ cm.


Then its total surface area $=\pi r^2+\pi rl=\dfrac{22}{7}\times 100+\dfrac{22}{7}\times 10\times 3.5=314.159+110=424.159$ cm$^2$.

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 of a right circular cone is $2 \,cm$ and its slant height is $3.5 \,cm$, then its curved surface area is

  1. $44 \,cm^2$
  2. $77 \,cm^2$
  3. $22 \,cm^2$
  4. $154 \,cm^2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given the radius of the base of a right circular cone $(r)=2$ cm, and its slant height $(l)=3.5$ cm.

Then its curved surface area $=\pi rl=\dfrac{22}{7}\times 2\times 3.5=22$ cm$^2$.

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

Diameter of the base of a cone is $10.5$cm and its slant height is $10$cm. Find its curved surface area.

  1. $104.85cm^2$
  2. $164.85cm^2$
  3. $100.75cm^2$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
diameter of box $=10.5\ cm$
radius $=\dfrac{10.5}{2}=5.25\ cm$
height $=l=10\ cm$
curved surface area $=(52.5 \pi)\ cm^{2}$
$\therefore$ Curved surface area of the given cone is $164.85\ cm^{2}$
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

If a right circular cone having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in ${ cm }^{ 2 }$ ) of this cone is

  1. $8\sqrt { 3\pi } $
  2. $6\sqrt { 2\pi } $
  3. $6\sqrt { 3\pi } $
  4. $8\sqrt { 2\pi } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a cone of maximum volume inscribed in a sphere of radius R, the height h = 4R/3 = 4 cm and radius r = sqrt(8/9) * R = sqrt(8) cm. The slant height l = sqrt(r^2 + h^2) = sqrt(8/9 * 9 + 16) = sqrt(8 + 16) = sqrt(24) = 2*sqrt(6). Curved surface area = pi * r * l = pi * sqrt(8) * 2*sqrt(6) = pi * 2*sqrt(2) * 2*sqrt(6) = 8*sqrt(3)*pi. The provided answer matches this form.

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

The $T.S.A$ of a cone whose $d=14\ cm$, $h=24\ cm$

  1. $504\ cm^{2}$
  2. $3696\ cm^{2}$
  3. $704\ cm^{2}$
  4. $528\ cm^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total surface area of a cone is given by pi * r * (l + r). Given diameter d = 14 cm, radius r = 7 cm, height h = 24 cm, the slant height l = sqrt(7^2 + 24^2) = 25 cm. Thus TSA = (22 / 7) * 7 * (25 + 7) = 22 * 32 = 704 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 a cone of radius $7$ cm and height $24$ cm is

  1. $440\ \text{cm}^2$
  2. $550\ \text{cm}^2$
  3. $330\ \text{cm}^2$
  4. $110\ \text{cm}^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Radius $r = 7 cm$

Height $h = 24 cm$

To find slant height $l$

$l _{2}^{2}=r^{2}+h^{2}$

$l^{2}=7^{2}+24^{2}$

$l^{2}=625$

$l=25 cm $

Curved surface area of cone is $\pi rl$

                               $=\dfrac{22}{7}\times 7\times 25$

                               $= 22\times 25$

                               $=550 cm ^{2}$

Curved surface area of cone is $550cm^{2}$
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} $