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 total area of sheet required to make an open cone of height 24 cm and radius 7 cm is

  1. 470 $ \displaystyle cm^{2} $
  2. 450 $ \displaystyle cm^{2} $
  3. 425 $ \displaystyle cm^{2} $
  4. 550 $ \displaystyle cm^{2} $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given height of cone is 24 cm and radius is 7 cm 

Then sheet required = surface area of cone =$\pi r(\sqrt{r^{2}+h^{2}})=2\times \frac{22}{7}\times 7(\sqrt{(24)^{2}+(7)^{2}})=2\times 22\times 25=550 cm^{2}$

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

If the diameter of a right cone is 6 cm and its vertical height is 4 cm then its curved surface area is

  1. 47.1 $ \displaystyle cm^{2} $
  2. 48 $ \displaystyle cm^{2} $
  3. 49 $ \displaystyle cm^{2} $
  4. 50 $ \displaystyle cm^{2} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given the vertical height is 4 cm and diameter is 6 cm

Then radius =$\frac{6}{2}=3$
And slant height =$l=\sqrt{r^{2}+h^{2}}=\sqrt{(3)^{2}+(4)^{2}}=\sqrt{9+16}=\sqrt{25}=5$ cm
Then curved surface area of cone =$\pi rL=\pi \times 3\times 5=3.14\times 5\times 3=47.1 cm^{2}$

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

If the area of the base of a right circular cone is 51 $ \displaystyle m^{2}     $and volume is 68 $ \displaystyle m^{3}     $ then its vertical height is

  1. 3.5 m

  2. 4 m

  3. 4.5 m

  4. 5 m
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the area of  base of right circular cone is 51 sq m and volume is 68 cu m

Then area of base =$\pi r^{2}=51$
$\Rightarrow r^{2}=\frac{51}{\pi }$
$\therefore volume =\frac{1}{3}\pi r^{2}h=68$
$\Rightarrow\frac{1}{3} \pi \left ( \frac{51}{\pi } \right )h= 68$
$\Rightarrow h=\frac{68}{17}$
$\Rightarrow h=4$ m

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

The radii of two right circular cone are int eh ratio of 4 : 5 and their slant heights are in the ratio 2:3 Then the ratio of their curved surfaces is

  1. 6 : 17

  2. 8 : 15

  3. 1 : 1

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

Let the radii of two right circular cone is 4x and 5x and siant height is 2y and 3y

Then curved surface area of first  right circular cone=$\pi \times 4x\times 2y=8\pi xy$ sq cm
And curved surface area of second right circular cone=$\pi \times 5x\times 3y=15\pi xy$ sq cm
So ratio of both right circular cone=$8\pi xy:15\pi xy::8:15$

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

The circumference of the base of a $24$ m high conical tent is $44$ m Calculate the length of canvas used in making the tent if the width of the canvas is $2$ m

  1. $257$ m
  2. $275$ m
  3. $752$ m
  4. $285$ m
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the radius of the base be r meters Then
$2$$\displaystyle \pi r$$=44$ m
or $\displaystyle r=\frac{44}{2\pi }=\frac{44\times 7}{2\times 22}=7 m$
Let the slant height be l meters Then
$\displaystyle l=\sqrt{h^{2}+r^{2}}=\sqrt{7^{2}+24^{2}}=\sqrt{625}=25m$
Now, Surface area = $\displaystyle \pi rl=\frac{22}{7}\times 7\times 25=550 m^{2}$
$\displaystyle \therefore $ Area of canvas required $= 550$ m$\displaystyle ^{2}$
 Given that width of the canvas is $2$ m therefore
Length of canvas required = $\displaystyle \frac{550}{2}=275m$

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

The slant height of a cone is $ 40$  m, the radius of the base is  $12$ m. Find the curved surface area of a cone.

  1. $\displaystyle 1507.2{ \ mm }^{ 2 }$
  2. $\displaystyle 1507.2\ m$
  3. $\displaystyle 1507.2{\ m }^{ 2 }$
  4. $\displaystyle 1407.2{\ mm }^{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the forrmula for Curved surface area of the cone$= \pi rs$

where, $\pi=3.14$
$radius(r)=12\ m$
$slant\ height(s)=40\ m$
$\therefore$ Curved surface area$=3.14 \times 12 \times 40=1507.2m^2$

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