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

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 
$\therefore \frac{4}{3}\pi r^{2}H=\pi r^{2}h\Rightarrow H=\frac{3h}{4}$

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

The canvas required to construct a cone of height $24$ m and base radius $7$ m is

  1. $500$ $ \displaystyle \ \text{m} ^{2} $
  2. $520$ $ \displaystyle \ \text{m} ^{2} $
  3. $550$ $ \displaystyle \ \text{m} ^{2} $

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given the height of cone 24 m and base radius is 24 m

Then slant height =$\sqrt{r^{2}+h^{2}}=\sqrt{(24)^{2}+(7)^{2}}=\sqrt{576+49}=\sqrt{625}=25$
Then canvas required to constrict con=the curved surface area of cone=$\frac{22}{7}\times 7\times 25=550 cm^{2}$

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

Find the surface area of a cone whose radius is $12$ m and the slant height is $23$ m.

  1. $\displaystyle 420\pi { \ mm }^{ 2 }$
  2. $\displaystyle 420\pi {\ cm }^{ 2 }$
  3. $\displaystyle 420\pi \ { m }^{ 2 }$
  4. $\displaystyle 420\pi \ m$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Surface area of cone is $A=πr(r+l)$

Here, radius is $r=12$ m and slant height is $l=23$ cm.

Thus,

$A=πr(r+l)=π\times 12(12+23)=12π\times 35=420π$ m$^2$

Hence, the surface area of the cone is $420π$ m$^2$.

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

Find the surface area of a cone whose radius is $20$ cm and the slant height is $0.3$ cm. (use $\displaystyle \pi =3.14$)

  1. $\displaystyle 1274.84\ m$
  2. $\displaystyle 1274.84{ \ mm }^{ 2 }$
  3. $\displaystyle 1274.84{ \ cm }^{ 2 }$
  4. $\displaystyle 1274.84\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, radius $=20$ cm and height $=0.$ cm

Surface area of a cone $=$ $\displaystyle \pi r\left( r+s \right) $
$\displaystyle =3.14\times 20\times \left( 20+0.3 \right) $
$\displaystyle =3.14\times 20\times 20.3$
$\displaystyle =1274.84{ cm }^{ 2 }$

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 is $\displaystyle 320{\  m }^{ 2 }$ whose radius is $7$ m. Find the surface area of a cone. (Assume $\displaystyle \pi =\frac { 22 }{ 7 } $)

  1. $\displaystyle 474{\ m }^{ 2 }$
  2. $\displaystyle 424{\ m }^{ 2 }$
  3. $\displaystyle 404{ \ m }^{ 2 }$
  4. $\displaystyle 454\ m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, surface area of cone $=20$ $m^2$ and radius $7$ m

Curved surface area of a cone $=$ $\displaystyle \pi rs$
$\displaystyle =320+\left( { 22 }/{ 7 } \right) \times 7\times 7$
$\displaystyle =320+154$
$\displaystyle =474{ m }^{ 2 }$

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

Calculate the surface area of a cone whose radius is $\dfrac{1}{3}$ cm and slant height is $12$ cm.

  1. ${3\pi}$ $cm^2$
  2. $\dfrac { 37\pi }{ 9 }$ $ { cm }^{ 2 }$
  3. $\dfrac { 37\pi }{ 8 }$ $ { cm }^{ 2 }$
  4. $\dfrac { 5\pi }{ 9 }$ $ { cm }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Surface area of cone is $A=πr(r+l)$


Here, radius is $r=\dfrac { 1 }{ 3 }$ cm and slant height is $l=12$ cm.

Thus,
 
$A=πr(r+l)=π\times \dfrac { 1 }{ 3 } \left( \dfrac { 1 }{ 3 } +12 \right) =\dfrac { 1 }{ 3 } π\times \dfrac { 37 }{ 3 } =\dfrac { 37π }{ 9 }$ cm$^2$
 
Hence, the surface area of the cone is $\dfrac { 37π }{ 9 }$ cm$^2$.

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

A cone has a radius of $2$ cm and height of $3$ cm, find total surface area of the cone.

  1. $\displaystyle 35.168\ cm$
  2. $\displaystyle 36.158{ \ cm }^{ 2 }$
  3. $\displaystyle 35.168{ cm }^{ 2 }$
  4. $\displaystyle 35.168{\ m }^{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

First need to find the value of slant height(s) of a cone, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 3 }^{ 2 }+{ 2 }^{ 2 }$
$\displaystyle { s }^{ 2 }=9+4$
$\displaystyle s=\sqrt { \left( 13 \right)  } $
$\displaystyle s=3.6$ cm
Then, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 2\times 3.6+3.14\times 2\times 2$
$\displaystyle =22.608+12.56$
$\displaystyle= 35.168{ cm }^{ 2 }$

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

A conical water tank has a radius of $0.2$ mm and height of $1.2$ mm, find total surface area of the tank.

  1. $\displaystyle 0.9734\ cm$
  2. $\displaystyle 0.9734{ \ cm }^{ 2 }$
  3. $\displaystyle 0.9734\ mm$
  4. $\displaystyle 0.9734{\ mm }^{ 2 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, radius $=0.2$ mm and height $=1.2$ mm 
Then find the value of slant height(s) of a conical water tank, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 1.2 }^{ 2 }+{ 0.2 }^{ 2 }$
$\displaystyle { s }^{ 2 }=1.44+0.4$
$\displaystyle s=\sqrt { \left( 1.84 \right)  } $
$\displaystyle s=1.35$ mm
So, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 0.2\times 1.35+3.14\times 0.2\times 0.2$
$\displaystyle =0.8478+0.1256$
$\displaystyle =0.9734{\  mm }^{ 2 }$

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

Calculate the surface area of cone whose curved surface area is $\displaystyle 100{ \ cm }^{ 2 }$ and its radius $200$ cm.

  1. $\displaystyle 125700\ cm$
  2. $\displaystyle 125700{\ mm }^{ 2 }$
  3. $\displaystyle 125700{\ m }^{ 2 }$
  4. $\displaystyle 125700{\ cm }^{ 2 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Surface area of a cone $= $ Curved surface area of a cone $+$ Area of circle
So, SA $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =100+3.14\times 200\times 200$
$\displaystyle =100+125600$
$\displaystyle =125700{ cm }^{ 2 }$

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

Find the surface area of a conical hat. If its slant height is three times the radius, the base diameter of a hat is $4$ inches. (Use $\displaystyle \pi =3$)

  1. $\displaystyle 48{ \ in }^{ 2 }$
  2. $\displaystyle 36\ in$
  3. $\displaystyle 48\ in$
  4. $\displaystyle 46{\ in }^{ 2 }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Surface area of cone is $A=πr(r+l)$


Here, the diameter is $4$ in and therefore, the radius is half of diameter that is $r=2$ in and it is also given that slant height is thrice the radius that is $l=(3\times 2)=6$ in. We use $π=3$.

Thus,
 
$A=πr(r+l)=3\times 2\left( 2+6 \right) =3\times 2\times 8=48$ in$^2$
 
Hence, the surface area of the conical hat is $48$ in$^2$.