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

The formula used for surface area of cone is  ($s$ denotes slant height.)

  1. $\displaystyle \pi r\left( r+s \right) $ sq.units
  2. $\displaystyle \pi r\left( 2r+s \right) $ sq.units
  3. $\displaystyle 2\pi r\left( r+s \right) $ sq.units
  4. $\displaystyle \pi r{ \left( r+s \right) }^{ 2 }$ sq.units
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Surface area of a cone $=$ $\displaystyle \pi r\left( r+s \right) $ sq.units
Since, surface area of cone $=$  Area of sector $+$ Area of circle.

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

The curved surface area of the cone is $\pi r l$ whereas, total surface area of the cone is

  1. $\pi r(r+l)$
  2. $\pi rl(r+l)$
  3. $2\pi rl(r-l)$
  4. $2\pi r^2l(r+l)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Surface area of cone $=$ CSA of cone + area of circular base

                                    $=$ $\pi rl+\pi { r }^{ 2 }\Rightarrow \pi r(l+r)$

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

If the circumference at the base of a right circular cone and the slant height are $120\pi$ $cm$ and $10cm$ respectively, then the curved surface area of the cone is equal to

  1. $1200\pi$ ${cm}^{2}$
  2. $600\pi$ ${cm}^{2}$
  3. $300\pi$ ${cm}^{2}$
  4. $600$ ${cm}^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have,

circumference $=120\pi\ cm$, slant height$(l)=10\ cm$
$\Rightarrow 2\pi r=120\pi$
$\Rightarrow 2r=120$
$\Rightarrow r=60\ cm$

We know that the curved surface area of the cone
$=\pi r l$

So,
$=\pi \times 60 \times 10$

$=600\pi\ cm^2$

Hence, this is the answer

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

Two right circular cones have equal radii. If their slant heights are in the ratio $4:3$, then their respective curved surface areas are in the ratio

  1. $16:9$
  2. $2:3$
  3. $4:3$
  4. $3:4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We have,

The radius of the two cone are equal

The ratio of the slant height $l _1:l _2=4:3$

We know that the curved surface area of the cone
$=\pi r l$

So, the required ratio

$=\dfrac{\pi r l _1}{\pi r l _2}$

$=\dfrac{l _1 }{l _2 }$

$=\dfrac{4}{3}=4:3$

Hence, this is the answer.

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 right circular cone of height $15$ cm and base diameter $16$ cm is __________.

  1. $60\pi \ \text{cm}^2$
  2. $68\pi \ \text{ cm}^2$
  3. $120\pi \ \text{cm}^2$
  4. $136\pi \ \text{cm}^2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The formula of curved surface area of a right circular cone $ =\pi \times r\times l $

$ \Rightarrow l=\sqrt { { (h }^{ 2 } } +{ r }^{ 2 }) $
$ \Rightarrow l=\sqrt { 8^{ 2 } } +{ 15 }^{ 2 }) $
$ \Rightarrow l=17 $ cm
Now substitute the value in above equation. we have
Curved surface area $ =\pi \times 8\times 17 $
$ \Rightarrow 136\pi \ { \text{cm} }^{ 2 } $
So, option D is the correct.

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 $616\, cm^2$ and its height is 48 cm. The total surface area of cone is 

  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
Given the area of the base of  a cone is $616\ cm^2$.
If $r$ be the radius of the base of the cone then 
$\pi r^2=616$
or, $\dfrac{22}{7}\times r^2=616$
or, $r=14$.
Also, given height $(h)=48\ cm$.
Then slant height $(l)=\sqrt{48^2+14^2}=2\sqrt{625}=50\ cm$.
$\therefore$ total surface area of the cone $=\pi r(r+l)=\dfrac{22}{7}\times 14\times 64=2816\ cm^2$.
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The base radii of a cone and a cylinder are equal. If their curved surface areas are also equal, then the ratio of the slant height of the cone to the height of the cylinder is

  1. $2 : 1$
  2. $1 : 2$
  3. $1 : 3$
  4. $3 : 1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let the radius of the cone and cylinder be $r$.
The base radii of cone and cylinder are equal.
Given curved surface areas are equal,
$\therefore (\pi)rl = 2(\pi)rh$
$\therefore \dfrac {l}{h}=2$
Hence, option A is correct.
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The height of a conical tent at the center is $5 m$, the distance of any point on its circular base from the top of the tent is $13 m$. The area of the slant surface is 

  1. $144 \pi sq. m$
  2. $130 \pi sq. m$
  3. $156 \pi sq. m$
  4. $169 \pi sq. m$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given Height $h=5$ and Slant height $l=13$


$\Rightarrow r^{2}=l^{2}-h^{2}=13^{2}-5^{2}$

$\Rightarrow r=12 $

Area of slant surface $=\pi rl$

                                    $=\pi ×12×13$

$\Rightarrow $area of slant surface $=156\pi$ sq.m

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 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 $550cm^{2}$