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

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 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

If the base radius and slant height of a right circular cone are $10 \,cm$ and $3.5 \,cm$ respectively, then its total surface area is

  1. $424.159 cm^2$
  2. $434.159 cm^2$
  3. $414.159 cm^2$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given the radius of the base of a right circular cone $(r)=10$ cm, and its slant height $(l)=3.5$ cm.


Then its total surface area $=\pi r^2+\pi rl=\dfrac{22}{7}\times 100+\dfrac{22}{7}\times 10\times 3.5=314.159+110=424.159$ cm$^2$.

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

If the radius of the base of a right circular cone is $2 \,cm$ and its slant height is $3.5 \,cm$, then its curved surface area is

  1. $44 \,cm^2$
  2. $77 \,cm^2$
  3. $22 \,cm^2$
  4. $154 \,cm^2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given the radius of the base of a right circular cone $(r)=2$ cm, and its slant height $(l)=3.5$ cm.

Then its curved surface area $=\pi rl=\dfrac{22}{7}\times 2\times 3.5=22$ cm$^2$.

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

Diameter of the base of a cone is $10.5$cm and its slant height is $10$cm. Find its curved surface area.

  1. $104.85cm^2$
  2. $164.85cm^2$
  3. $100.75cm^2$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
diameter of box $=10.5\ cm$
radius $=\dfrac{10.5}{2}=5.25\ cm$
height $=l=10\ cm$
curved surface area $=(52.5 \pi)\ cm^{2}$
$\therefore$ Curved surface area of the given cone is $164.85\ cm^{2}$
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

If a right circular cone having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in ${ cm }^{ 2 }$ ) of this cone is

  1. $8\sqrt { 3\pi } $
  2. $6\sqrt { 2\pi } $
  3. $6\sqrt { 3\pi } $
  4. $8\sqrt { 2\pi } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a cone of maximum volume inscribed in a sphere of radius R, the height h = 4R/3 = 4 cm and radius r = sqrt(8/9) * R = sqrt(8) cm. The slant height l = sqrt(r^2 + h^2) = sqrt(8/9 * 9 + 16) = sqrt(8 + 16) = sqrt(24) = 2*sqrt(6). Curved surface area = pi * r * l = pi * sqrt(8) * 2*sqrt(6) = pi * 2*sqrt(2) * 2*sqrt(6) = 8*sqrt(3)*pi. The provided answer matches this form.

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

  1. $440\ \text{cm}^2$
  2. $550\ \text{cm}^2$
  3. $330\ \text{cm}^2$
  4. $110\ \text{cm}^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Radius $r = 7 cm$

Height $h = 24 cm$

To find slant height $l$

$l _{2}^{2}=r^{2}+h^{2}$

$l^{2}=7^{2}+24^{2}$

$l^{2}=625$

$l=25 cm $

Curved surface area of cone is $\pi rl$

                               $=\dfrac{22}{7}\times 7\times 25$

                               $= 22\times 25$

                               $=550 cm ^{2}$

Curved surface area of cone is $550cm^{2}$
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Mark the correct alternative of the following.
A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to that of the cone is?

  1. $3:5$
  2. $2:5$
  3. $3:1$
  4. $1:3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

It is given that, the volumes of both cylinder and cone are the same.

So, let volume of the cylinder and cone be $V.$
It is also given that, their base radii are the same.
So, let radius of the cylinder $=$ Radius of the cone $=r$
Let the height of the cylinder and the cone be $h _1$ and $h _2$ respectively.
Volume of cone $=\dfrac{1}{3}\pi r^2 h _2$
Volume of cylinder $=\pi r^2 h _1$
We know both volumes are same.
$\therefore$  $\dfrac{1}{3}\pi r^2 h _2=\pi r^2 h _1$

$\Rightarrow$  $\dfrac{h _1}{h _2}=\dfrac{\pi r^2}{3\pi r^2}$

$\therefore$  $\dfrac{h _1}{h _2}=\dfrac{1}{3}$

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

Mark the correct alternative of the following.
If the base radius and the height of a right circular cone are increased by $20\%$, then the percentage increase in volume is approximately.

  1. $60$
  2. $68$
  3. $73$
  4. $78$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let original radius of base and height are $R$ and $H$ respectively.
Original height $=H$

New radius $=\dfrac{120}{100}R=\dfrac{6}{5}R$

And new height $=\dfrac{6}{5}H$

Original volume of cone $V _1=\dfrac{1}{3}\pi R^2 H$

New volume of cone $V _2=\dfrac{1}{3}\pi \left(\dfrac{6}{5}R\right)^2\times \dfrac{6}{5}H$

                                          $=\dfrac{216}{125}V _1$

Now, increased in volume $=\dfrac{216}{125}V _1-V _1=\dfrac{91}{125}V _1$

$\therefore$  Percentage increased in volume $=\left(\dfrac{91}{125}V _1\times\dfrac{1}{V _1}\right)\times 100\%$

                                                              $=72.81\%\approx 73\%$

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

Mark the correct alternative of the following.
The total surface area of a cone of radius $\dfrac{r}{2}$ and length $2l$, is?

  1. $2\pi r(l+r)$
  2. $\pi r\left(1+\dfrac{r}{4}\right)$
  3. $\pi r(l+r)$
  4. $2\pi rl$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $r$ and $l$ be base radius and slant height of cone.

Total surface area $=\pi r (l+r)$
Here, it is given that, 
The base radius is $\dfrac{r}{2}$ and that the slant height is $2l.$
Substituting these values in the above equation we have,
Total surface area $=\pi\left(\dfrac{r}{2}\right)\left(2l+\dfrac{r}{2}\right)$

                               $=\pi r\left(l+\dfrac{r}{4}\right)$

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

If the radius of the base and the height of a right circular cone are respectively $21$ cm and $28$ cm, then the curved surface area of the cone is $\displaystyle \left(\pi\, =\, \frac{22}{7}\right)$

  1. $3696\, cm^{2}$
  2. $2310\, cm^{2}$
  3. $2550\, cm^{2}$
  4. $2410\, cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a cone, $l = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $ where h is the height. 
Hence, $l = \sqrt { { 28 }^{ 2 }+  {21}^{ 2 } } $

$ l = 35 $ cm 

Curved surface area of a cone $= \pi rl$  where r is the radius of the cone and $l$ is the slant height.
Hence, curved surface area of this cone $ = \dfrac {22}{7} \times 21 \times 35 = 2310  {cm}^{2} $

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

If the radius and slant height of a cone are in the ratio $4 : 7$ and its curved surface area is $792 cm^{2}$, then its radius is $\displaystyle \left(\pi\, =\, \frac{22}{7}\right)$.

  1. $10$ cm
  2. $8$ cm
  3. $12$ cm
  4. $9$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the radius of the cone be $ 4a$ and slant height $ = 7a$ 

Given, curved surface area $=792$ $cm^2$
Curved surface area of a cone $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.

Hence, curved surface area of this cone $ = \displaystyle \frac {22}{7} \times 4a \times 7a = 792 $
$ \therefore  a = 3 $ cm 

Hence, radius $ = 4a  = 12 $ cm