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 area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

The volume, V $cm^{2}$, of a hollow cylindrical pipe of length $l$ cm, outer radius R cm and inner radius r cm is given by the formula : $V\, =\, \pi\, (R^{2}\, -\, r^{2}).\, l$

Find r, if $V\, =\, 22,\, R\, =\, 2,\, l\, =\, 4$ and $\pi,\, 3\displaystyle \frac{1}{7}.$

  1. 1.5

  2. 1.2

  3. 1.4

  4. 1.6
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $V= \pi \left ( R^{2}-r^{2} \right )l$

$V= \pi  R^{2}-\pi r^{2} l$

$\pi r^{2}l= \pi R^{2}l-V$


$ r^{2}= \dfrac{\pi R^{2}l-V}{\pi l}$

$\therefore  r= \sqrt{\dfrac{\pi R^{2}l-V}{\pi l}}$

Given $V=22 ,R=2 ,L=4 , \pi = 3\tfrac{1}{7}= \frac{22}{7}$

$\therefore r= \sqrt{\dfrac{\frac{22}{7}\times 4\times 4-22}{\dfrac{22}{7}\times4}}= \sqrt{\dfrac{352-154}{88}}= \sqrt{\dfrac{198}{88}}= \sqrt{\dfrac{9}{4}}= 1.5$

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

The diameters of two cylinders are in the ratio of 2:1 and their volumes are equal. The ratio of their heights will be _________.

  1. 1:6

  2. 1:2

  3. 1:4

  4. 3:4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let the diameter of the given two cylinders are $2x$ and $x$, 
so the radius of these cylinders are $x$ and $0.5 x$ respectively. 

Let the height of these cylinders are $ h _1$ and $ h _2$ respectively.

Given $ πx^2h _1=π(0.5x)^2h _2 $

$∴\dfrac{h _1}{h _2}=\dfrac{(0.5)^2}{1}=\dfrac 14$

$1:4.$
Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

If the volume of a cylinder is $448\pi:cm^3$ and height 7 cm, its total surface area will be ______________.

  1. $352\:cm^2$
  2. $754.28\:cm^2$
  3. $724.64\:cm^2$
  4. $354\:cm^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let the radius of cylinder is $r$ cm and height of this is $7$ cm.

Given, volume of cylinder
$=πr^2h$

$448π=πr^2×7⟹r2=64⟹r=8$ cm

∴  Required total surface area of cylinder
$=2πr(r+h)=2π×8(8+7)$

$=2×\dfrac {22} 7×8×15 $

$=754.28$ $cm^2$
Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

A rectangular paper of dimensions 6 cm and 3 cm is rolled to form a cylinder with height equal to the width of the paper, then its base radius is

  1. $ \displaystyle \frac{6}{\pi }cm $
  2. $ \displaystyle \frac{3}{2\pi }cm $
  3. $ \displaystyle \frac{6}{2\pi }cm $
  4. $ \displaystyle \frac{9}{2\pi }cm $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The length of the rectangle become the circumference of the base of the cylinder 

$\therefore 2\pi r=6\Rightarrow r=\frac{6}{2\pi }$ cm

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

The curved surface of a circular cylinder of height 'h' and the curved surface area of the cone of slant height 2 'h' having the same circular base are in the ratio of

  1. 1 : 2

  2. 2 : 1

  3. 1 : 1

  4. 1 : 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the base radius of cone and circular cylinder is r and height of circular cylinder is h and height of cone 2h

Then Curved surface area of circular cylinder =$2\pi rh$
And curved surface area of cone=$\pi r(2h)=2\pi rh$
So ratio of Curved surface area of circular cylinder : curved surface area of cone :: $2\pi r(h)=2\pi rh$ : $2\pi rh$=1:1

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

The height of a hollow cylinder is $14cm$ if external diameter is $16cm$ and total curved surface area of the hollow cylinder is $1320sq.cm$, then its internal diameter is

  1. $14cm$
  2. $16cm$
  3. $7cm$
  4. $8cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given     

external radius $r _2=8$, height of cylinder $h=14$
 we have,


$2\pi h(r _{1}+r _{2})=1320$

$ \implies8+r _1=\displaystyle \frac{1320\times7}{2\times22\times14}$

$\implies r _1=7cm$

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

The ratio between the radius of the base and the height of a cylinder is $2:3$. If its volume is $12936$ cu. cm, the total  surface area of the cylinder is :

  1. $2587.2 c{m^2}$
  2. $3080 c{m^2}$
  3. $25872 c{m^2}$
  4. $38808 c{m^2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
We have $\dfrac{r}{h}=\dfrac{2}{3}\Rightarrow\,h=\dfrac{3r}{2}$

Volume of a cylinder$=\pi{r}^{2}h$

$\Rightarrow\,12936=\dfrac{22}{7}\times{r}^{2}\times \dfrac{3r}{2}$

$\Rightarrow\,12936=\dfrac{11\times 3}{7}{r}^{3}$

$\Rightarrow\,{r}^{3}=\dfrac{12936\times 7}{33}=2744$

$\Rightarrow\,r=\sqrt[3]{2744}=14\ cm$

We have $h=\dfrac{3r}{2}=\dfrac{3\times 14}{2}=21\ cm$

Total Surface area$=2\pi\,r\left(r+h\right)=2\times\dfrac{22}{7}\times 14\left(14+21\right)=2\times\dfrac{22}{7}\times 14\times 35=140\times 22=3080\ sq.cm$
Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

A cylinder and cone of equal base radius and equal height are given. Which of the following statement is true/

  1. Volume of cylinder and cone are equal

  2. Volume of cylinder is one-third of volume of cone

  3. Volume of cone is half of the volume of cylinder

  4. Volume of cone is one-third of volume of cylinder
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The volume of a cylinder is given by V_cyl = pi r^2 h, while the volume of a cone with the same base radius and height is V_cone = (1/3) pi r^2 h. Therefore, the volume of the cone is exactly one-third of the volume of the cylinder.

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

A cylindrical pipe is made from a metal sheet of length 88 cm and breadth 20 cm. What is the volume of this pipe?

  1. $2800$ ${ cm }^{ 3 }$
  2. $12320$ ${ cm }^{ 3 }$
  3. $13202$ ${ cm }^{ 3 }$
  4. $13220$ ${ cm }^{ 3 }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We have,

Length $l=88\,cm.$

Breadth$b=20\,cm.$

Volume $=?$

Volume of cylindrical pipe $=\pi {{r}^{2}}h$

We know that,

$ circumfrance=Breadth=2\pi r $

$ 2\pi r=20 $

$ \pi r=10 $

$ r=\dfrac{10}{\pi } $

$ r=\dfrac{10}{\dfrac{22}{7}} $

$ r=\dfrac{70}{22} $

$ r=\dfrac{35}{11}\,\,cm. $

Then,

Volume of cylindrical pipe $V=\pi {{r}^{2}}h$

$ V=\dfrac{22}{7}\times \dfrac{35}{11}\times \dfrac{35}{11}\times 88 $

$ V=2\times 5\times 35\times 8 $

$ V=80\times 35 $

$ V=2800\,c{{m}^{3}} $

Hence, this is the answer.

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

If sum of radius and height of a cylinder is 6, then its maximum volume is 

  1. $32\pi$
  2. $16\pi$
  3. $8\pi$

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

Volume V = pi * r^2 * h. Given r + h = 6, then h = 6 - r. V(r) = pi * r^2 * (6 - r) = pi * (6r^2 - r^3). To maximize, V'(r) = pi * (12r - 3r^2) = 0. So 3r(4 - r) = 0, r = 4. Then h = 2. Max Volume = pi * 4^2 * 2 = 32 * pi.

Multiple choice maths area and volume of cylinder hollow cylinder volume of cylinder volume of cylinder and cone

Mark the correct alternative of the following.
In a cylinder, if radius is doubled and height is halved, curved surface area will be?

  1. Halved

  2. Doubled

  3. Same

  4. Four times
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the radius of cylinder be $r$ and height be $h.$

So, the original curved surface area $=2\pi rh$
When, radius is doubled and height is halved,
New curved surface area $=2\pi \times 2r\times \dfrac{h}{2}$

                                           $=2\pi r h$
$\therefore$  New curved surface area $=$ Original surface area.
$\therefore$  There is no change in the curved surface area of the cylinder