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 circumference of the base of a 9 m high wooden solid cone is 44 m The slant height of the cone is

  1. $\displaystyle \sqrt{110}m$
  2. $\displaystyle \sqrt{130}m$
  3. $\displaystyle \sqrt{150}m$
  4. $\displaystyle \sqrt{180}m$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let r be the radius of the base of cone.

So, circumference of the base of the cone$=2\pi r=44  m$
$\Rightarrow 2\times \dfrac{22}{7}\times r=44$    $[\because \pi=\dfrac{22}{7}]$
$\Rightarrow r=\dfrac{44\times 7}{22\times 2}=7  m$
So the radius of the cone=7 m
$\therefore  Slant  height =\sqrt{r^2+h^2}$
$\Rightarrow \sqrt{7^2+9^2}=\sqrt{49+81}=\sqrt{130}  m$


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

The radius of a conical tent is 12 m and the slant height is 5.6 m Find the length of canvas required to make the tent it the width of canvas is 4 m

  1. 106.6 m

  2. 100 m

  3. 52.8 m

  4. 105.6 m
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given r=12 m l=5.6 m B=4 m
Total surface area of the tent=Area of the canvas
i.e. $\displaystyle \pi rl=L\times B$
or $\displaystyle \frac{22}{7}\times 12\times 5.6=L\times 4$
or 211.2=4L
or $\displaystyle L=\frac{211.2}{4}=52.8m$'

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 slant height l and radius r is given by

  1. $\frac {1}{3}\pi /r^2$
  2. $\pi rl$
  3. $\pi rl^2$
  4. $\frac {1}{3}\pi rl$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Consider the given slant height l and radius$ =r$ of the cone,

 Curved surface area=(Arc length of sector/Circumference of circle)×Area of circle Curved surface area

$=\dfrac{2\pi r}{2\pi l}\times \pi {{l}^{2}}=\pi rl$ 




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 volume of the cone whose vertical height is 8 m and the area of base $ \displaystyle 156  m^{2}     $ is

  1. 416 $ \displaystyle m^{2} $
  2. 415 $ \displaystyle m^{2} $
  3. 312 $ \displaystyle m^{2} $
  4. 468 $ \displaystyle m^{2} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given the vertical height of cone is 8 m and area of base is 156 sq m

Let radius of cone is r m 
Then base area =$\pi r^{2}=156\Rightarrow r^{2}=156 \pi  m$
And the volume of cone=$\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times 156\pi \times 8=416 m^{3}$

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

If the area of the base of a right circular cone is 51 $ \displaystyle m^{2}     $and volume is 68 $ \displaystyle m^{3}     $ then its vertical height is

  1. 3.5 m

  2. 4 m

  3. 4.5 m

  4. 5 m
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the area of  base of right circular cone is 51 sq m and volume is 68 cu m

Then area of base =$\pi r^{2}=51$
$\Rightarrow r^{2}=\frac{51}{\pi }$
$\therefore volume =\frac{1}{3}\pi r^{2}h=68$
$\Rightarrow\frac{1}{3} \pi \left ( \frac{51}{\pi } \right )h= 68$
$\Rightarrow h=\frac{68}{17}$
$\Rightarrow h=4$ m

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 circumference of the base of a $24$ m high conical tent is $44$ m Calculate the length of canvas used in making the tent if the width of the canvas is $2$ m

  1. $257$ m
  2. $275$ m
  3. $752$ m
  4. $285$ m
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the radius of the base be r meters Then
$2$$\displaystyle \pi r$$=44$ m
or $\displaystyle r=\frac{44}{2\pi }=\frac{44\times 7}{2\times 22}=7 m$
Let the slant height be l meters Then
$\displaystyle l=\sqrt{h^{2}+r^{2}}=\sqrt{7^{2}+24^{2}}=\sqrt{625}=25m$
Now, Surface area = $\displaystyle \pi rl=\frac{22}{7}\times 7\times 25=550 m^{2}$
$\displaystyle \therefore $ Area of canvas required $= 550$ m$\displaystyle ^{2}$
 Given that width of the canvas is $2$ m therefore
Length of canvas required = $\displaystyle \frac{550}{2}=275m$

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$=3.14 \times 12 \times 40=1507.2m^2$