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

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
$\Rightarrow 940=3.14 \times 1\times 4s+3.14\times 14^2$
$\Rightarrow 940=43.96\times s+3.14\times 196$
$\Rightarrow 940=43.96\times s+615.44$
$\Rightarrow 940-615.44=43.96\times s$
$\Rightarrow 324.56=43.96\times s$
$\Rightarrow s=\dfrac{324.56}{43.96}$
$\Rightarrow s=7.38\approx 7\ m$

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

A closed cone tank radius $7$ cm and height $10$ cm is made from a sheet of aluminium. How much sheet is required?

  1. $144cm^2$
  2. $\displaystyle 22\sqrt { 149 } cm^2$
  3. $\displaystyle (22\sqrt { 149 } +144) cm^2$

  4. None of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total surface area of cone  = Area of base + Area of curved surface
$\displaystyle =\quad \pi { r }^{ 2 }+\pi rs$

$\displaystyle =\frac { 22 }{ 7 } \times 7\times 7+\frac { 22 }{ 7 } \times 7\left( \sqrt { 149 }  \right) $
$\displaystyle s=\sqrt { { h }^{ 2 }+{ r }^{ 2 } } $
$\displaystyle =\sqrt { 149+100 } $
$\displaystyle =144+22\sqrt { 149 } $

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

Find  the radius of the base of a right circular cone which has a lateral surface area of $6\pi$ and a slant height of $6$ ( in standard units )

  1. $0.50$
  2. $0.75$
  3. $1.00$
  4. $1.25$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, Lateral surface area $=6 \pi$ and slant height $=6$

Let the radius of base be $r$ and slant height of cone be $l = 6$.
Lateral surface area is equal to $\pi rl = \pi \times r \times 6 = 6\pi$
$\Rightarrow 6 \pi= \pi \times r \times 6$
$\Rightarrow r=1$

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

Find the slant height and vertical height of a Cone with radius $5.6$ cm and curved surface area $158.4$ cm$^2$.

  1. $8.07$
  2. $7.05$
  3. $8$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Radius $=5.6$ cm, vertical height $= h$, slant height $=$ $l$
Curved Surface Area of cone $=\pi r l = 158.4 cm^2$
$\Rightarrow \dfrac{22}{7} \times 5.6 \times l = 158.4$
$\Rightarrow l = \dfrac{158.4 \times 7}{22 \times 5.6} = \dfrac{18}{2} = 9$ cm
We know $l^2 = r^2 + h^2$
Thus $h^2 = l^2 - r^2 $

$= 9^2 - (5.6)^2$
$= 81 - 31.36$
$= 49.64$
$h = \sqrt{49.64}$
$h = 7.05 $ cm (approx.)

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

The curved surface area of right circular cone with height $24$ m and radius $7$ m is

  1. $500\ \text{m}^{2}$
  2. $550\ \text{m}^{ 2 }$
  3. $607\ \text{m}^{ 2 }$
  4. $650\ \text{m}^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Height of cone$=24m$
Radius of cone$=7m$
Slant height of cone$=\sqrt { { 24 }^{ 2 }+{ 7 }^{ 2 } } =\sqrt { 576+49 } =\sqrt { 625 } =25m$
CSA of cone$=\pi rl=\cfrac { 22 }{ 7 } \times 7\times 25=550 m^2$
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A cone and a cylinder have the same base area. They also have the same curved surface area. If the height of the cylinder is $3$ m, then the slant height of the cone (in m) is

  1. $3$
  2. $4$
  3. $6$
  4. $7$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given the radius of cylinder and cone are the same because there base areas are same.
Curved surface of cylinder $=$ curved surface area of the cone
$\therefore 2 \pi r h = \pi r l$
$\therefore l = 2h $
Given, height of cylinder $= 3$ cm
Therefore, slant height of cone $= 6$ cm

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 with height $24$ cm and radius $7$ cm is

  1. $500 cm^2$
  2. $550 cm^2$
  3. $607 cm^2$
  4. $650 cm^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given,
Radius of cone $= 7$ cm
Height of cone $= 24$ cm
Curved surface area = $\pi r \sqrt{(r^2 + l^2)}$
= $\pi \times 7 (\sqrt{(7^2 + (24)^2)}$
= $\pi \times 70\times 25$
= $550 cm^2$

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

Consider two cones, the curved surface area of one being twice that of the other and the slant height of the later being twice that of the former. The ratio of the radius of the later cone to that of the former is

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

Let the curved surface areas of two cones be $C _1$ and $C _2$

And their slant heights be $l _1$ and $l _2$
Given, $C _1=2C _2$
$l _2=2l _1$

$\therefore \pi r _1l _1=2\pi r _2.2l _1$
$\therefore r _1=4r _2$