Quantitative Aptitude

Mensuration

2,507 Questions
All Direct Formula Application 1067 Shape Transformations 343 Mensuration Ratio Equations 277 Combined & Hollow Solids 247 Successive Percentage Changes 178 Inscribed & Circumscribed Figures 165 Cost Estimation Problems 100 Liquid Pour & Submergence 67 Unit Conversions 35 Path & Border Calculations 17 Time & Work Integration 7 Scale & Model Proportions 2

Mensuration Questions

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

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 cone is $\displaystyle 286 m^{2}$ and its the slant height is 13 m, then volume is

  1. $\displaystyle 821.389m^{3}$
  2. $\displaystyle 852.258m^{3}$
  3. $\displaystyle 364.369m^{3}$
  4. $\displaystyle 281.164m^{3}$
Reveal answer Fill a bubble to check yourself
D 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.
Hence, CSA of this cone, $ = \frac {22}{7} \times r \times 13 = 286 $
$ => r = 7  m $

For a cone, l $ = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $ where lis the slant height.

Hence, $ 13 = \sqrt { { h }^{ 2 }+  {7}^{ 2 } } $ 

$ 169 = { h }^{ 2 } + 49 $

$ { h }^{ 2 } = 120 $

$ h = 2 \sqrt {30}  m $

Hence, volume of this cone $ = \frac { 1 }{ 3 } \times \frac { 22 }{ 7 } \times { 7 }^{ 2 }\times 2 \sqrt {30} = \frac {154}{3} \sqrt {30} { m }^{ 3 } $