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 similarity areas of similar figures areas of similar triangles relations between the areas of triangles

Let $\displaystyle \Delta XYZ$ be right angle triangle with right angle at Z. Let $\displaystyle A _{X}$ denotes the area of the circle with diameter YZ. Let $\displaystyle A _{Y}$ denote the area of the circle with diameter XZ and let $\displaystyle A _{Z}$ denotes the area of the circle diameter XY. Which of the following relations is true?

  1. $\displaystyle A _{Z}=A _{X}+A _{Y}$
  2. $\displaystyle A _{Z}=A^{2} _{X}+A^{2} _{Y}$
  3. $\displaystyle A^{2} _{Z}=A^{2} _{X}+A^{2} _{Y}$
  4. $\displaystyle A^{2} _{Z}=A^{2} _{X}-A^{2} _{Y}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In $\triangle XYZ$, using Pythagoras theorem,
$XY^2 = XZ^2 + YZ^2$
$\pi XY^2 = \pi XZ^2 + \pi YZ^2$ (Multiply by $\pi$)
$A _z = A _x + A _y$

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 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 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

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 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 =$\pi rl=\pi \times 6\times 10==60\pi cm^{2}$

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

The total area of sheet required to make an open cone of height 24 cm and radius 7 cm is

  1. 470 $ \displaystyle cm^{2} $
  2. 450 $ \displaystyle cm^{2} $
  3. 425 $ \displaystyle cm^{2} $
  4. 550 $ \displaystyle cm^{2} $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given height of cone is 24 cm and radius is 7 cm 

Then sheet required = surface area of cone =$\pi r(\sqrt{r^{2}+h^{2}})=2\times \frac{22}{7}\times 7(\sqrt{(24)^{2}+(7)^{2}})=2\times 22\times 25=550 cm^{2}$

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

If the diameter of a right cone is 6 cm and its vertical height is 4 cm then its curved surface area is

  1. 47.1 $ \displaystyle cm^{2} $
  2. 48 $ \displaystyle cm^{2} $
  3. 49 $ \displaystyle cm^{2} $
  4. 50 $ \displaystyle cm^{2} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given the vertical height is 4 cm and diameter is 6 cm

Then radius =$\frac{6}{2}=3$
And slant height =$l=\sqrt{r^{2}+h^{2}}=\sqrt{(3)^{2}+(4)^{2}}=\sqrt{9+16}=\sqrt{25}=5$ cm
Then curved surface area of cone =$\pi rL=\pi \times 3\times 5=3.14\times 5\times 3=47.1 cm^{2}$

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

The radii of two right circular cone are int eh ratio of 4 : 5 and their slant heights are in the ratio 2:3 Then the ratio of their curved surfaces is

  1. 6 : 17

  2. 8 : 15

  3. 1 : 1

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the radii of two right circular cone is 4x and 5x and siant height is 2y and 3y

Then curved surface area of first  right circular cone=$\pi \times 4x\times 2y=8\pi xy$ sq cm
And curved surface area of second right circular cone=$\pi \times 5x\times 3y=15\pi xy$ sq cm
So ratio of both right circular cone=$8\pi xy:15\pi xy::8:15$

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

The slant height of a cone is $ 40$  m, the radius of the base is  $12$ m. Find the curved surface area of a cone.

  1. $\displaystyle 1507.2{ \ mm }^{ 2 }$
  2. $\displaystyle 1507.2\ m$
  3. $\displaystyle 1507.2{\ m }^{ 2 }$
  4. $\displaystyle 1407.2{\ mm }^{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the forrmula for Curved surface area of the cone$= \pi rs$

where, $\pi=3.14$
$radius(r)=12\ m$
$slant\ height(s)=40\ m$
$\therefore$ Curved surface area$=3.14 \times 12 \times 40=1507.2m^2$