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 lateral surface area (in ${cm}^{2}$) of a cone with height $3$ cm and radius $4$ cm is:

  1. $62\cfrac{6}{7}$
  2. $52\cfrac{6}{7}$
  3. $31\cfrac{3}{7}$
  4. $15\cfrac{5}{7}$
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.
For a cone,  $l = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $, where $h$ is the height.

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

$ l = \sqrt {25} $

$ l = 5 $ cm

Hence, Curved surface area of this cone, $ =\displaystyle  \frac {22}{7} \times 4 \times 5 = 62 \frac {6}{7}  {cm}^{2} $

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

Find the total surface area of a cone, if its slant height is $9\ m$ and the radius of its base is $12\ m$.

  1. $792\ {m}^{2}$
  2. $452\ {m}^{2}$
  3. $682\ {m}^{2}$
  4. $987\ {m}^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total surface area of a cone $ = \pi r (r + l) $  

Hence, TSA of this cone, $ = \cfrac {22}{7} \times 12 \times (12 + 9) = 792  {m}^{2} $

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

The diameter of a cone is $14\ cm$ and its slant height is $9\ cm$. Find the area of its curved surface.

  1. $198\ {cm}^{2}$
  2. $108\ {cm}^{2}$
  3. $152\ {cm}^{2}$
  4. $218\ {cm}^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

Radius of the cone $ = \cfrac {Diameter}{2} = \cfrac {14}{2}  =  7  cm $

Hence, CSA of this cone $ = \cfrac {22}{7} \times 7 \times 9 = 198  {cm}^{2} $

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

Slant height of a cone is 13 cm and radius is 7 cm its lateral surface area is

  1. 280 $\displaystyle cm^{2}$
  2. 282 $\displaystyle cm^{2}$
  3. 284 $\displaystyle cm^{2}$
  4. 286 $\displaystyle cm^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle L.S.A\, of\, a\, cone =\pi rl$
$\displaystyle =\dfrac{22}{7}\times 7\times 13=286cm^{2}$

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 10 m high conical tent is 44 metres Then the length of canvas used in making the tent if width of canvas is 2 m is (Use $\displaystyle \pi =22/7$)

  1. $132.2 m$
  2. $134.2 m$
  3. $130.2 m$
  4. $136.2 m$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let r m be the radius of the base h m be the height and l m be the slant height of the cone Then 
Circumference=44 metres
$\displaystyle \Rightarrow 2\pi r=44\Rightarrow 2\times \frac{22}{7}\times r=44\Rightarrow r=7$
metres It is given that h=10 metres
$\displaystyle \therefore l^{2}=r^{2}+h^{2}\Rightarrow l=\sqrt{r^{2}+h^{2}}$
$\displaystyle=\sqrt{49+100}=\sqrt{149}=12.2m$
Now surface area of the tent $\displaystyle =\pi rl$
$\displaystyle \frac{22}{7}\times 7\times 12.2m^{2}=268.4m^{2}$
$\displaystyle \therefore $ Area of the canvas used $\displaystyle =268.4m^{2}$
It is given that the width of the canvas is 2 m
$\displaystyle \therefore $ Length of the canvas used=$\displaystyle =\frac{area}{width}=\frac{268.4}{2}=134.2m$
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The volume of a right circular cone of height $8 cm$ and radius of base $3 cm$ is

  1. $12 \pi cm^3$
  2. $24 \pi cm^3$
  3. $48 \pi cm^3$
  4. $72 \pi cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given that height of right circular cone $h=8cm$

Radius of right circular cone $r=3cm$

Volume of right circular cone$V=\dfrac{1}{3}{{r}^{2}}\pi .h$

$ =\dfrac{1}{3}{{.3}^{2.}}\pi .8c{{m}^{3}} $

$ =24\pi c{{m}^{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 circumference of the base of a 9 m high wooden solid cone is 44 m The slant height of the cone is

  1. $\displaystyle \sqrt{110}m$
  2. $\displaystyle \sqrt{130}m$
  3. $\displaystyle \sqrt{150}m$
  4. $\displaystyle \sqrt{180}m$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let r be the radius of the base of cone.

So, circumference of the base of the cone$=2\pi r=44  m$
$\Rightarrow 2\times \dfrac{22}{7}\times r=44$    $[\because \pi=\dfrac{22}{7}]$
$\Rightarrow r=\dfrac{44\times 7}{22\times 2}=7  m$
So the radius of the cone=7 m
$\therefore  Slant  height =\sqrt{r^2+h^2}$
$\Rightarrow \sqrt{7^2+9^2}=\sqrt{49+81}=\sqrt{130}  m$


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

The total surface area of a cone is $\displaystyle 770cm^{2}$ If its slant height is four times the radius of the cone the diameter fo the cone is

  1. 14 cm

  2. 7 cm

  3. 20 cm

  4. 18 cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total surface area of cone $\displaystyle TSA _{cone}=770cm^{2}$
$\displaystyle l=4r$
$\displaystyle \therefore TSA _{cone}=\pi rl+\pi r^{2}=\pi r\times 4r+\pi r^{2}$
or $\displaystyle 770=5\pi r^{2}$
or $\displaystyle r^{2}=\frac{770}{5\times 22}\times 7=49$
$\displaystyle \therefore r=\sqrt{49}=7$
$\displaystyle \therefore$ Diameter, d=$\displaystyle 2\times 7$=14 cm

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

The radius and the slant height of a cone are in the ratio 7:13 and the curved surface area is $\displaystyle 286cm^{2}$ Find its radius

  1. 5 cm

  2. 7 cm

  3. 8.8 cm

  4. 10.3 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $\displaystyle \frac{r}{l}=\frac{7}{13}=x$
Curved surface area CSA=286 $\displaystyle cm^{2}$
But CSA=$\displaystyle \pi rl$
$\displaystyle \therefore 286=\frac{22}{7}\times 7x\times 13x$
or $\displaystyle x^{2}=\frac{286}{22\times 13}=1$
or x=1
$\displaystyle \therefore $ r=7 cm
$\displaystyle \therefore $ Diameter d=14 cm

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

The radius of a conical tent is 12 m and the slant height is 5.6 m Find the length of canvas required to make the tent it the width of canvas is 4 m

  1. 106.6 m

  2. 100 m

  3. 52.8 m

  4. 105.6 m
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given r=12 m l=5.6 m B=4 m
Total surface area of the tent=Area of the canvas
i.e. $\displaystyle \pi rl=L\times B$
or $\displaystyle \frac{22}{7}\times 12\times 5.6=L\times 4$
or 211.2=4L
or $\displaystyle L=\frac{211.2}{4}=52.8m$'

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

Consider the given slant height l and radius$ =r$ of the cone,

 Curved surface area=(Arc length of sector/Circumference of circle)×Area of circle Curved surface area

$=\dfrac{2\pi r}{2\pi l}\times \pi {{l}^{2}}=\pi rl$ 




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

If a sphere has the same curved surface area as total surface area of cone of vertical height 40 cm and radius 30 cm then the radius of the sphere is

  1. $ \displaystyle 10\sqrt{6} $ cm
  2. $ \displaystyle 10\sqrt{3} $ cm
  3. $ \displaystyle 10\sqrt{2} $ cm

  4. 12 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the vertical height of cone is 40 cm and radius is 30 cm 

Then  surface area of cone=$\pi rh=\pi \times 30\times 40=1200 \pi cm^{2}$
The surface area of cone =covered surface area of sphere
$\therefore \pi R^{2}=1200 \pi \Rightarrow R^{2}=300\Rightarrow R=10\sqrt{3}cm$
Then the radius of sphere =$10\sqrt{3}cm$

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

The volume of the cone whose vertical height is 8 m and the area of base $ \displaystyle 156  m^{2}     $ is

  1. 416 $ \displaystyle m^{2} $
  2. 415 $ \displaystyle m^{2} $
  3. 312 $ \displaystyle m^{2} $
  4. 468 $ \displaystyle m^{2} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given the vertical height of cone is 8 m and area of base is 156 sq m

Let radius of cone is r m 
Then base area =$\pi r^{2}=156\Rightarrow r^{2}=156 \pi  m$
And the volume of cone=$\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times 156\pi \times 8=416 m^{3}$

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 84 cm and diameter 70 cm is

  1. 10010 $ \displaystyle cm^{2} $
  2. 100000 $ \displaystyle cm^{2} $
  3. 10020 $ \displaystyle cm^{2} $
  4. 11000 $ \displaystyle cm^{2} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given 84 cm is the height of right circular cone and diameter is 70 cm 

Then radius of  right circular cone=$\frac{70}{2}=35 cm$
Then curved surface area of a right circular cone =$\pi r(\sqrt{r^{2}+h^{2}})=\frac{22}{7}\times 35\left ( \sqrt{(35)^{2}+(84)^{2}} \right )=110\left ( \sqrt{7056+1225} \right )=110\times 91=10010$ sq cm

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

The slant height of a right circular cone is 10 m and its height is 8 cm then the area of its curved surface is

  1. 80 $ \displaystyle \pi m^{2} $
  2. 60 $ \displaystyle \pi m^{2} $
  3. 65 $ \displaystyle \pi m^{2} $
  4. 70 $ \displaystyle \pi m^{2} $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given that slant height of a right circular cone is 10 cm and its height is 8 cm 

Then radius of cone $R^{2}=(10)^{2}-(8)^{2}=100-64=36\Rightarrow R=6 cm$
Then curved surface area of right circular cone =$\pi rl=\pi \times 6\times 10==60\pi cm^{2}$