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 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$^2$.

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 radius $14$ m and slant height $49$ m. (Use $\displaystyle \pi =\frac { 22 }{ 7 } $).

  1. $\displaystyle 2762$ $\ m^2$
  2. $\displaystyle 2772{ \ m }^{ 2 }$
  3. $\displaystyle 1772{\ m }^{ 2 }$
  4. $\displaystyle 2672{\ m }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given $r=14m,s=49m$
Surface area of a cone 
$=$ $\displaystyle \pi rs+\pi { r }^{ 2 }$

$\displaystyle =\dfrac {22}{7} \times 14\times 49+\dfrac {22}{7} \times 14\times 14$

$\displaystyle =2156+616$

$\displaystyle =2772{ m }^{ 2 }$

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

The total surface area of a conical jar is $\displaystyle 740{ ft }^{ 2 }$. If its slant height is two times the radius, then what is the base diameter of the colical jar? (use $\displaystyle \pi =3$).

  1. $18.12$ ft
  2. $18.10$ ft
  3. $18.24$ ft
  4. $18.31$ ft
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Formula:

Surface area of cone=$\pi rs+\pi r^2$
$s=2r$
where 
$r$ is the radius of the base of the cone.
$s$ is the slant height of the cone.
We know that surface area =$740\ ft^2$
$\therefore \pi rs+\pi r^2=740$
Substituting $s=2r$ and $\pi=3$ in the above equation we get,
$ 3 \times r \times 2r+3 r^2=740$
$\Rightarrow 3*2r^2+3r^2=740$
$\Rightarrow 6r^2+3r^2=740$
$\Rightarrow 9r^2=740$
$\Rightarrow r^2=\dfrac{740}{9}$
$\Rightarrow r^2=82.22$
$\Rightarrow r=\sqrt{82.22}$
$\Rightarrow r=9.06$
The diameter(d)=twice of radius
$\therefore d=2r$
$\therefore d=2 \times 9.06$
$\therefore d=18.12\ ft$

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

What is the total surface area of a cone if its diameter =${ 1}$ ft and slant height = $12$ ft. (use $\displaystyle \pi =3.14$).

  1. $\displaystyle 11.625{ ft }^{ 2 }$
  2. $\displaystyle 18.625{ ft }^{ 2 }$
  3. $\displaystyle 19.625{ ft }^{ 2 }$
  4. $\displaystyle 19.625ft$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle Diameter=\frac { radius }{ 2 } $
$\displaystyle radius=\dfrac{1}{2}ft$
Surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle 3.14\times { 1 }/{ 2 }\times 12+3.14\times { 1 }/{ 2 }\times { 1 }/{ 2 }$
$\displaystyle 18.84+0.785$
$\displaystyle 19.625{\  ft }^{ 2 }$

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

An ice cream cone has radius of $21$ m and height of $20$ m. Find total surface area of the cone.

  1. $\displaystyle 3297{\ m }^{ 2 }$
  2. $\displaystyle 9734{ \ cm }^{ 2 }$
  3. $\displaystyle 3297\ mm$
  4. $\displaystyle 3297\ m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Find the value of slant height(s) of a ice-cream cone, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 20 }^{ 2 }+{ 21 }^{ 2 }$
$\displaystyle { s }^{ 2 }=400+441$
$\displaystyle s=\sqrt { 841 } $
$\displaystyle s=29$ m
So, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 21\times 29+3.14\times 21\times 21$
$\displaystyle =1912.26+1384.74$
$\displaystyle =3297{ m }^{ 2 }$

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

What is the total surface area of a cone, if its radius = 5 cm and height = $\displaystyle\sqrt2 $cm?

  1. $\displaystyle 159.983{ \ mm }^{ 2 }$
  2. $\displaystyle 159.983{\ cm }^{ 2 }$
  3. $\displaystyle 159.983\ cm$
  4. $\displaystyle 159.983{ \ m }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The value of slant height(s) of a cone, using Pythagoras theorem, since the cross section is a right triangle is

$s^2=h^2+r^2$
where 
h= height of cone
r=radius of the base of the cone
we know that
$h=\sqrt2\ cm$
$r=5\ cm$
$\therefore s^2=\sqrt2^2+5^2=2+25=27$
$\Rightarrow s^2=27$
$\Rightarrow s=\sqrt27$
$\Rightarrow s=5.19\ cm$

Total Surface area of cone$=\pi rs+\pi r^2$
$\pi=3.14$
$r=5\ cm$
$s=5.19\ cm$
$\therefore \pi rs+\pi r^2=3.14\times 5\times5.19+3.14\times 5\times5$
$=81.483+78.5$
$=159.983\ cm^2$

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

The total surface area of a conical tent is $ 920$ square meter and its radius $14$ m. Find the slant height. (Round off your answer to the nearest whole number).

  1. $7$ m
  2. $6$ m
  3. $5$ m
  4. $6.5$ m
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Formula:

Surface area of cone$=\pi rs+\pi r^2$
where r is the radius of the base of the cone and s is the slant height.
We know that surface area of cone$=940\ m^2$
$r =14\ m$
$\pi=3.14$
Substituting the values in the formula we get
$\Rightarrow 940=3.14 \times 1\times 4s+3.14\times 14^2$
$\Rightarrow 940=43.96\times s+3.14\times 196$
$\Rightarrow 940=43.96\times s+615.44$
$\Rightarrow 940-615.44=43.96\times s$
$\Rightarrow 324.56=43.96\times s$
$\Rightarrow s=\dfrac{324.56}{43.96}$
$\Rightarrow s=7.38\approx 7\ m$