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Volume of cylinder - class-IX

Description: volume of cylinder
Number of Questions: 53
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Tags: mensuration mensuration - iii area and volume of cylinder cylinder, cone and sphere surface areas and volumes surface area and volume maths perimeter, area and volume
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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 $\pi,\, 3\displaystyle \frac{1}{7}.$

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

  1. 1.5

  2. 1.2

  3. 1.4

  4. 1.6

Show answer
Correct Option: A
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 $V=22 ,R=2 ,L=4 , \pi = 3\tfrac{1}{7}= \frac{22}{7}$

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

An iron pipe $20\space cm$ long has exterior diameter equal to $25\space cm$. If the thickness of the pipe is $1\space cm$, find the whole surface area of the pipe.

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

  1. $3167\space cm^2$

  2. $3160\space cm^2$

  3. $3068\space cm^2$

  4. $3268\space cm^2$

Show answer
Correct Option: A
Explanation:

TSA of pipe $=$ $2\pi h(R+r)+2\pi ({ R }^{ 2 }-{ r }^{ 2 })$


                     $=$ $2\times \dfrac { 22 }{ 7 } \times 20(12.5+11.5)+2\times \dfrac { 22 }{ 7 } \left( { \left( 12.5 \right)  }^{ 2 }-{ \left( 11.5 \right)  }^{ 2 } \right) $


                     $=$ $\dfrac { 44\times 480 }{ 7 } +\dfrac { 44\times 24 }{ 7 } $

                    $ =$ $\dfrac { 21120 }{ 7 } +\dfrac { 1056 }{ 7 } =\dfrac { 22176 }{ 7 } $

                     $=$ $3167$ ${ cm }^{ 2 }$

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

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

  1. 1:6

  2. 1:2

  3. 1:4

  4. 3:4

Show answer
Correct Option: C
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 $

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

$1:4.$

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

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

  1. $352:cm^2$

  2. $754.28:cm^2$

  3. $724.64:cm^2$

  4. $354:cm^2$

Show answer
Correct Option: B
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 $

$=754.28$ $cm^2$

200 wooden balls each of diameter 70 mm are to be painted Find the cost of painting these balls at 10 paise/$\displaystyle cm^{2}$

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

  1. Rs.3080

  2. Rs.2771

  3. Rs.4000

  4. Rs.7000

Show answer
Correct Option: A
Explanation:

Total surface area of 200 balls of $\displaystyle \frac{35}{10}$ cm radius will be $\displaystyle 200\times 4\times \frac{22}{7}\times \frac{35}{10}\times \frac{35}{10}$ mm
The cost will be Rs $\displaystyle \frac{200\times 4\times 22\times 35\times 35}{7\times 10\times 10\times 100}$ which simplifies to Rs 3080

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

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

  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 $

Show answer
Correct Option: C
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

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

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

  1. 1 : 2

  2. 2 : 1

  3. 1 : 1

  4. 1 : 3

Show answer
Correct Option: C
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

A hollow sphere of internal and external radii 3 cm and 4 cm respectively is malted into a cylinder of diameter 37 cm The height of the cylinder is

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

  1. 2 cm

  2. 2.5 cm

  3. 3 cm

  4. none

Show answer
Correct Option: D
Explanation:

Given the external  radius of hollow sphere is 4 cm and internal radius 3 cm 

Then volume of metal used=$\frac{4}{2}\pi (R^{2}-r^{2})=\frac{4}{3}\pi \left ( (4)^{2}-(3)^{2} \right )=\frac{4}{3}\pi (16-9)=\frac{4}{3}7\pi $
The diameter of cylinder 37 cm 
Radius of cylinder is =18.5 cm
Volume of cylinder=$\pi r^{2}h=\pi (18.5)^{2}h$
But cylinder made by metal of hollow sphere
$\pi (18.5\times 18.5)h=\frac{4}{3}7\pi \Rightarrow h=\frac{4\times 7}{3\times 18.5\times 18.5}\Rightarrow \Rightarrow h=0.027 cm$

A cistern $6$ m long and $4$ m wide contains water to a depth of $1.25$ m. What is the area of wetted surface?

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

  1. $40$ sq. m

  2. $45$ sq. m

  3. $49$ sq. m

  4. $73$ sq. m

Show answer
Correct Option: D
Explanation:

Given, $l = 6, b = 4$

Also given depth i.e., $ h = 1.25$
Area of the wetted surface $= 2[lb + bh + hl]$
$=2[(6\times 4)+(4\times 1.25)+(1.25\times 6)]$
$=2[24+5+7.5]$
$= 73$ sq. m
Therefore, the area of wetted surface is $73$ sq. m.

The outer and inner diameters of a circular pipe are $6$ cm and $4$ cm respectively. If its length is $10$ cm then what is the total surface area in square centimetres?

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

  1. $55\pi$

  2. $110\pi$

  3. $150\pi$

  4. None of the above

Show answer
Correct Option: D
Explanation:

Given, outer and inner diameters of circular pipe are $6$ cm and $4$ cm
Therefore, outer and inner radii of a circular pipe are $3$ cm and $2$ cm.
Thus total surface area would be $ 10\times \pi (3^{2} - 2^{2})$ $= 50\pi $ sq. cm.

What will be the volume of a sphere of diameter $15\ cm$? (Correct up to $2$ decimal place)

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

  1. $1436.76\ {cm}^{3}$

  2. $1767.15\ {cm}^{3}$

  3. $14137.17\ {cm}^{3}$

  4. $4188.79\ {cm}^{3}$

Show answer
Correct Option: B
Explanation:

The Volume of a sphere$=\dfrac{4}{3}\pi \ \text{R}^{3}$
Here Radius $R=7.5 \ cm$
On solving We get Volume$=1767.15\ {cm}^{3}$ 

What will be the radius of a sphere whose surface area is 616 $cm^{2}$? (Use $\pi=\dfrac{22}{7}$)

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

  1. $6$ cm

  2. $7$ cm

  3. $8$ cm

  4. $9$ cm

Show answer
Correct Option: B
Explanation:

Radius $=\dfrac{1}{2}\times\sqrt{\dfrac{\text{surface area}}{\pi}}$
on solving we get radius $=7$ cm

If volume of sphere is $850$ $m^{3}$ then its radius and surface area are

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

  1. $6m$, $450$ $m^{2}$

  2. $5m$, $560$ $m^{2}$

  3. $2m$, $780$ $m^{2}$

  4. $5.88m$, $434$ $m^{2}$

Show answer
Correct Option: D
Explanation:
Volume of Sphere $=850m^3=\cfrac{4}{3}\pi r^3 \Rightarrow r^3=\cfrac{850\times 3\times 7}{4\times 22}=202.84 \\ \Rightarrow r=\sqrt[3]{202.84}=5.88m$
Surface area $=4\pi r^2=4\times \cfrac{22}{7}\times 5.88\times 5.88 \approx 434m^2$

 If the radius of a sphere is doubled, then what is the ratio of new to the old surface area?

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

  1. $1:2$

  2. $2:1$

  3. $1:4$

  4. $4:1$

Show answer
Correct Option: D
Explanation:

$S _1=4\pi r^2 \ S _2=4\pi (2r)^2=16\pi r^2 \ \cfrac{S _2}{S _1}=\cfrac{16\pi r^2}{4\pi r^2}=\cfrac{4}{1} \Rightarrow 4:1$

A test-tube consists of a hollow cylindrical tube joined to a hemi-spherical bown of the same internal radius. The whole tube holds $350$ cc of water and in the cylindrical portion falls $1$ cm if $19.64$ cc of water is removed. Find the length of the cylindrical portion of the tube. (Take $\pi =$ $22/7$)

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

  1. $12.15$ cm

  2. $16.15 $ cm

  3. $24.15 $ cm

  4. None of these

Show answer
Correct Option: B
Explanation:

Let r cm. be the radius of the hemisphere and h cm be the length of the cylindrical portion.
Volume of water removed $\pi r^2 (1) = 19.64cc$
$\Rightarrow r^2 = 19.64 \times \displaystyle \frac{7}{22}  \Rightarrow r = 2.5 cm$
Volume of the whole tube $= \pi r^2 h + \displaystyle \frac{2}{h} \pi r^3 = 350 c.c.$
$\Rightarrow \pi r^2 \displaystyle \left ( h + \frac{2}{3} r \right ) = 350$
$\Rightarrow \displaystyle \frac{22}{7} \times 2.5^2 \times \left ( h + \frac{2}{3} \times 2.5 \right ) = 350$
$\displaystyle \frac{22}{7} \times 6.25 \times (h+ 1.67) = 350 $
$ \Rightarrow \displaystyle 350 \times \frac{7}{22} \times 6. 25 - 1.67 cm    \Rightarrow h = 16.15 cm$

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

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

  1. $14cm$

  2. $16cm$

  3. $7cm$

  4. $8cm$

Show answer
Correct Option: C
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}$

$\implies r _1=7cm$

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 :

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

  1. $2587.2 c{m^2}$

  2. $3080 c{m^2}$

  3. $25872 c{m^2}$

  4. $38808 c{m^2}$

Show answer
Correct Option: B
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$=2\pi\,r\left(r+h\right)=2\times\dfrac{22}{7}\times 14\left(14+21\right)=2\times\dfrac{22}{7}\times 14\times 35=140\times 22=3080\ sq.cm$

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

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

  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

Show answer
Correct Option: A

Find the volume of a solid cylinder whose radius is $14$cm and height $30$cm

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

  1. $18380cm^3$

  2. $18480cm^3$

  3. $18580cm^3$

  4. $18680cm^3$

Show answer
Correct Option: A

the radii of two cylinders are in the ratio 2:3 and their height are in the ratio 5:3. ratio of their volume

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

  1. $20:27$

  2. $10:9$

  3. $18:13$

  4. $9:20$

Show answer
Correct Option: A

A solid cylinder has a total surface area of $231cm^2$. Its curved surface area. Find the volume of the cylinder?

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

  1. $270cm^3$

  2. $269.5cm^3$

  3. $256.5cm^3$

  4. $289.5cm^3$

Show answer
Correct Option: A

Volume of a cylinder when $d=7\ cm$ and $h=3\ cm$

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

  1. $118\ cm^{3}$

  2. $115.5\ cm^{3}$

  3. $155.5\ cm^{3}$

  4. $808.5\ cm^{3}$

Show answer
Correct Option: A

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

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

  1. $2800$ ${ cm }^{ 3 }$

  2. $12320$ ${ cm }^{ 3 }$

  3. $13202$ ${ cm }^{ 3 }$

  4. $13220$ ${ cm }^{ 3 }$

Show answer
Correct Option: A
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.

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

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

  1. $32\pi$

  2. $16\pi$

  3. $8\pi$

  4. None of these

Show answer
Correct Option: A

Mark the correct alternative of the following.
Two cylindrical jars have their diameters in the ratio $3:1$, but height $1:3$. Then the ratio of their volumes is?

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

  1. $1:4$

  2. $1:3$

  3. $3:1$

  4. $2:5$

Show answer
Correct Option: C
Explanation:

Let $V _1$ and $V _2$ be the volume of the two cylinders with radius $r _1$ and height $h _1$, and radius $r _2$ and height $h _2.$

$\dfrac{2r _1}{2r _2}=\dfrac{3}{1}$ and $\dfrac{h _1}{h _2}=\dfrac{1}{3}$             [ Given ]
So,
$V _1=\pi r _1^2h _1$           ----- ( 1 )
Now,
$V _2=\pi r _2^2h _2$            ---- ( 2 )
From equation ( 1 ) and ( 2 ), we get
$\dfrac{V _1}{V _2}=\left(\dfrac{r _1}{r _2}\right)^2\left(\dfrac{h _1}{h _2}\right)$

$\Rightarrow$  $\dfrac{V _1}{V _2}=\left(\dfrac{2r _1}{2r _2}\right)^2\left(\dfrac{h _1}{h _2}\right)$

$\Rightarrow$  $\dfrac{V _1}{V _2}=(3)^2\left(\dfrac{1}{3}\right)$

$\Rightarrow$  $\dfrac{V _1}{V _2}=\dfrac{3}{1}$

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

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

  1. Halved

  2. Doubled

  3. Same

  4. Four times

Show answer
Correct Option: C
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

Mark the correct alternative of the following.
The radius of a wire is decreased to one-third. If volume remains the same, the length will become?

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

  1. $3$ times

  2. $6$ times

  3. $9$ times

  4. $27$ times

Show answer
Correct Option: C
Explanation:

Let $V _1$ and $V _2$ be the volume of the two cylinders with $h _1$ and $h _2$ as their heights.

Let $r _1$ and $r _2$ be their base radius.
It is given that, the radius of a wire is decreased to on-third.
$\therefore$  $r _2=\dfrac{1}{3}r _1$

$\Rightarrow$  $V _1=V _2$             [ Given ]
$\Rightarrow$  $\pi r _1^2 h _1=\pi r _2^2 h _2$

$\Rightarrow$  $r _1^2 h _1=\left(\dfrac{1}{3}r _1\right)^2 h _2$

$\Rightarrow$  $r _1^2 h _1=\dfrac{1}{9} r _1^2 h _2$

$\Rightarrow$  $h _2=9h _1$

$\therefore$  The length will become $9$ times.

Mark the correct alternative of the following.
If the height of a cylinder is doubled and radius remains the same, then volume will be?

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

  1. Doubled

  2. Halved

  3. Same

  4. Four times

Show answer
Correct Option: A
Explanation:

Let $V _1$ be the volume of the cylinder with radis $r _1$ and height $h _1,$ then

$\Rightarrow$  $V _1=\pi r _1^2 h _1$            ---- ( 1 )
Now, let $V _2$ be the volume after changing the dimension, then
$\Rightarrow$  $r _2=r _1,$  $h _2=2h _1$
So,
$\Rightarrow$  $V _2=\pi r _2^2h _2$

$\Rightarrow$  $V _2=\pi\times{r _1}^2\times 2h _1$

$\Rightarrow$  $V _2=2\times \pi r _1^2 h _1$
From ( 1 ),

$\Rightarrow$  $V _2=2V _1$

$\therefore$  If the height of a cylinder is doubled and radius remains the same, then volume will be $Doubled.$

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

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

  1. Same

  2. Doubled

  3. Halved

  4. Four times

Show answer
Correct Option: C
Explanation:

Let $V _1$ be the volume of the cylinder with radis $r _1$ and height $h _1,$ then

$\Rightarrow$  $V _1=\pi r _1^2 h _1$            ---- ( 1 )
Now, let $V _2$ be the volume after changing the dimension, then
$\Rightarrow$  $r _2=\dfrac{1}{2}r _1,$  $h _2=2h _1$
So,
$\Rightarrow$  $V _2=\pi r _2^2h _2$

$\Rightarrow$  $V _2=\pi\times\left(\dfrac{r _1}{2}\right)^2\times 2h _1$

$\Rightarrow$  $V _2=\dfrac{1}{2}\times \pi r _1^2 h _1$
From ( 1 ),

$\Rightarrow$  $V _2=\dfrac{1}{2}V _1$

$\therefore$  In a cylinder, if radius is halved and height is doubled, the volume will be $Halved$

Mark the correct alternative of the following.
If the radius of a cylinder is doubled and the height remains same, the volume will be?

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

  1. Doubled

  2. Halved

  3. Same

  4. Four times

Show answer
Correct Option: D
Explanation:

Let $V _1$ be the volume of the cylinder with radius $r _1$ and height $h _1,$ then

$V _1=\pi r _1^2 h _1$           ---- ( 1 )
Now, let $V _2$ be the volume after changing the dimensions, then
$r _2=2r _1,\,h _2=h _1$
So,
$\Rightarrow$  $V _2=\pi r _2^2 h _2$

$\Rightarrow$  $V _2=\pi\times(2r _1)^2\times h _1$

$\Rightarrow$  $V _2=4\times \pi r _1^2 h _1$
From ( 1 ),
$\therefore$  $V _2=4V _1$
Hence, If the radius of a cylinder is doubled and the height remains same, the volume will be Four times.

Mark the correct alternative of the following.
The volume of a cylinder of radius r is $1/4$ of the volume of a rectangular box with a square base of side length x. If the cylinder and the box have equal heights, what is r in terms of x?

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

  1. $\dfrac{x^2}{2\pi}$

  2. $\dfrac{x}{2\sqrt{\pi}}$

  3. $\dfrac{\sqrt{2x}}{\pi}$

  4. $\dfrac{\pi}{2\sqrt{x}}$

Show answer
Correct Option: B
Explanation:

Let the height of the cylinder be $h.$

Volume of the cylinder $=\pi r^2 h$
Height of the rectangular box $=h$
Since, base is square with side $x.$
Volume of the box $=x\times x\times h=x^2 h$
According to question,
$\Rightarrow$  $\pi r^2  h=\dfrac{1}{4} x^2 h$

$\Rightarrow$  $r^2=\dfrac{1}{4\pi}x^2$
Taking square root on both sides,
$\Rightarrow$  $r=\dfrac{x}{2\sqrt{\pi}}$

Mark the correct alternative of the following.
Two circular cylinders of equal volume have their heights in the ratio $1:2$. Ratio of their radii is?

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

  1. $1:\sqrt{2}$

  2. $\sqrt{2}:1$

  3. $1:2$

  4. $1:4$

Show answer
Correct Option: A
Explanation:
$\dfrac{h _1}{h _2}=\dfrac{1}{2}$        [ Given ]
Let $V _1$ and $V _2$ are volume of cylinders.

$\therefore$  $V _1=V _2$          [ Given ]

$\therefore$  $\dfrac{V _1}{V _2}=1$

$\Rightarrow$  $\dfrac{\pi r _1^2h _1}{\pi r _2^2 h _2}=1$

$\Rightarrow$  $\left(\dfrac{r _1}{r _2}\right)^2\left(\dfrac{h _1}{h _2}\right)=1$

But it is given that,
$\dfrac{h _1}{h _2}=\dfrac{1}{2}$

$\therefore$  $\left(\dfrac{r _1}{r _2}\right)^2\times\dfrac{1}{2}=1$

$\Rightarrow$  $\left(\dfrac{r _1}{r _2}\right)^2=2$

$\Rightarrow$  $\left(\dfrac{r _1}{r _2}\right)^2=\dfrac{2}{1}$

$\Rightarrow$  $\dfrac{r _1}{r _2}=\dfrac{\sqrt{2}}{1}$

$\therefore$  The ratio of the radii of the two cylinders is $\sqrt{2}:1$

Mark the correct alternative of the following.
The altitude of a right circular cylinder is increased six times and the base area is decreased one-ninth of its value. The factor by which the lateral surface of the cylinder increases, is?

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

  1. $\dfrac{2}{3}$

  2. $\dfrac{1}{2}$

  3. $\dfrac{3}{2}$

  4. $2$

Show answer
Correct Option: D
Explanation:

Curved surface area of cylinder $=2\pi r h$


Height is increased to $6$ times $=6h$


Base area is decreased to $\left(\dfrac{1}{9}th\right)$ 

i.e. $\pi (r{^{\prime}})^2=\dfrac{1}{9}\pi r^2\Rightarrow r^{\prime}=\dfrac13r $

Now,
New curved surface area $=2\pi\times\dfrac{1}{3}r\times 6h$

                                           $=2\times(2\pi r h)$
$\therefore$  Lateral surface area becomes twice.

Mark the correct alternative of the following.
The height h of a cylinder equal the circumference of the cylinder. In terms of h, what is the volume of the cylinder?

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

  1. $\dfrac{h^3}{4\pi}$

  2. $\dfrac{h^2}{2\pi}$

  3. $\dfrac{h^3}{2}$

  4. $\pi h^3$

Show answer
Correct Option: A
Explanation:

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

It is given that,
$2\pi r =h$
$\Rightarrow$  $r=\dfrac{h}{2\pi}$
Therefore, the volume of the cylinder is
$V=\pi r^2 h$

$\Rightarrow$  $V=\pi\left(\dfrac{h}{2\pi}\right)^2h$

$\Rightarrow$  $V=\dfrac{h^3}{4\pi}$

Mark the correct alternative of the following.
If the heights of two cones are in the ratio of $1:4$ and the radii of their bases are in the ratio $4:1$, then the ratio of their volumes is?

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

  1. $1:2$

  2. $2:3$

  3. $3:4$

  4. $4:1$

Show answer
Correct Option: D
Explanation:

The base radius of cone is $'r'$ and vertical height $'h'$.

$\Rightarrow$  Volume of cone $=\dfrac{1}{3}\pi r^2 h$
Let the base radius and height of the two cones be $r _1,h _1$ and $r _2,h _2$ respectively.
It is given that the ratio between the heights of the two cones is $1:4$.
Since, only the ratio is given, to use them in our equation we introduce a constant $'k'.$
So,
$h _1=1k$
$h _2=4k$
It is also given that, the ratio between the base radius of the two cones is $4:1.$
Since, only the ratio is given, to use then in our equation we introduce another constant $'p'$
So,
$r _1=4p$
$r _2=1p$
Let $V _1$ and $V _2$ be the volumes of cones.

$\Rightarrow$  $\dfrac{V _1}{V _2}=\dfrac{\pi\times 4p\times 4p\times 1k\times 3}{3\times \pi\times 1p\times 1p\times 4k}$

$\therefore$   $\dfrac{V _1}{V _2}=\dfrac{4}{1}$

A hollow cylindrical pipe is $21 \ cm$ long. If its outer and inner diameters are $10 \ cm$ and $6 \ cm$ respectively, them the volume of the metal used in making the pipe is $\displaystyle \left(Take\, \pi\, =\, \frac{22}{7}\right)$

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

  1. $1048\, cm^{3}$

  2. $1056\, cm^{3}$

  3. $1060\, cm^{3}$

  4. $1064\, cm^{3}$

Show answer
Correct Option: B
Explanation:

The pipe is in the shape of a hollow cylinder.
Volume of a hollow Cylinder of outer Radius "R", inner Radius ""r" and height "h" $ = \pi ({ R }^{ 2 }-{ r }^{ 2 })h $
Outer Radius $ = \frac {10}{2} = 5  cm $
Inner Radius $ = \frac {6}{2} = 3  cm $
Hence, volume of the pipe $ = \frac { 22 }{ 7 } \times ({ 5 }^{ 2 }-{ 3 }^{ 2 })\times 21 = 1056  {cm}^{3} $

If the volume of a vessel in the form of a right circular cylinder is 448 $\pi\, cm^{3}$ and its height is 7 cm, then the curved surface area of the cylinder is

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

  1. $224\, \pi\, cm^{2}$

  2. $212\, \pi\, cm^{2}$

  3. $112\, \pi\, cm^{2}$

  4. None of these

Show answer
Correct Option: C
Explanation:

Volume of a Cylinder of Radius $R$ and height $h$ $ = \pi { R }^{ 2 }h $
$\therefore $ volume of the given cylinder $ =\pi \times {R}^{2} \times 7  = 448 \pi  {cm}^{3} $

$ {R}^{2} = 64 $

$ R = 8 cm $  

Curved surface area of a cylinder of radius "$R$" and height "$h$" $ = 2\pi Rh$

$\therefore$ curved surface area of the given cylinder $ = 2\times \pi \times 8\times 7 =  112 \pi   \  cm^2 $

A hollow iron pipe of $21 cm$ long and its external diameter is $8 cm$. If the thickness of the pipes is $1 cm$ and iron weights $\displaystyle 8g/cm^{2}$, then the weight of the pipe is equal to

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

  1. $3.6 kg$

  2. $3.696 kg$

  3. $36 kg$

  4. $36.9 kg$

Show answer
Correct Option: B
Explanation:
Let external diameter be ${ d } _{ 2 }$ and internal be ${ d } _{ 1 }$
$\Rightarrow { d } _{ 2 }=8cm$ & ${ d } _{ 1 }=8-1-1=6cm$
Therefore, ${ r } _{ 1 }=3cm$ & ${ r } _{ 2 }=4cm$
Volume of hollow cylindrical pipe$=\pi \left( { r } _{ 2 }^{ 2 }-{ r } _{ 1 }^{ 2 } \right) \times h$
$=\cfrac { 22 }{ 7 } \left( { \left( 4 \right)  }^{ 2 }-{ \left( 3 \right)  }^{ 2 } \right) \times 21$
$=\cfrac { 22 }{ 7 } \times 7\times 21$
$=462cm^3$
$\therefore $Weight of the pipe$=8\times 462=3696g=3.696kg$

A rectangular sheet of width $14$ m is rolled along its width and is converted to form a cylinder. Find the radius of cylinder.

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

  1. $\displaystyle \frac { 22 }{ 49 } $

  2. $\displaystyle \frac { 44 }{ 29 } $

  3. $\displaystyle \frac { 49 }{ 22 } $

  4. None

Show answer
Correct Option: C
Explanation:

Curved surface area of cylinder$=100m^2$

$\therefore 2 \pi r h = 100$
Here, width of the rectangle = height of the cylinder.
$\therefore h=14m$

$\therefore 2 \times \dfrac {22}{7} \times r \times 14 = 100$

$ \therefore r = \dfrac {100 \times 7}{2 \times 22 \times 14}$

In a cylinder, if the radius is halved and height is doubled, the curved surface area will 

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

  1. remain same

  2. increase

  3. decrease

  4. none of the above

Show answer
Correct Option: A
Explanation:

Volume of cylinder $\displaystyle \pi { r }^{ 2 }h$
Now, if $\displaystyle r=\frac { r }{ 2 } & h=h2$
New CSA $\displaystyle =2\pi rh$
$\displaystyle =2\pi \left( \frac { r }{ 2 }  \right) \times \left( h\times 2 \right) $
$\displaystyle =2\pi rh$
It will remain same

The circumference of base of cylindrical reservoir is $\displaystyle 30\pi cm$ and height is $10$ cm. How many litres of water can it hold?

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

  1. $\displaystyle 2.1\pi$ litres

  2. $\displaystyle 2.25\pi$ litres

  3. $\displaystyle 225\pi$ litres

  4. $\displaystyle 2250\pi$ litres

Show answer
Correct Option: B
Explanation:

Circumference $\displaystyle 2\pi r=30\pi ,r=\frac { 30\pi  }{ 2\pi  } =15$

Volume of cylinder $\displaystyle =\pi { r }^{ 2 }h$
$\displaystyle =\left( \pi \times 10\times 15\times 15 \right) { cm }^{ 3 }$
$\displaystyle =2250\pi { cm }^{ 3 }$
$\displaystyle =2.25\pi l$

The inner diameter of a circular well is $3.5$ m. It is $10$ m deep. Find the cost of plastering this curved surface at the rate of Rs. $40$ per m$^2$.

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

  1. Rs. $4000$

  2. Rs. $4400$

  3. Rs. $4500$

  4. Rs. $4800$

Show answer
Correct Option: B
Explanation:

Given diameter of well is $3.5$ m and depth is $10$ m and cost of plastering is Rs. $40$ per sq m.

Then radius of well $=\dfrac{3.5}{2}=1.75$ m
And height of well $=10$ m
Then curved surface area of well $=$ $2\pi rh=2\times \dfrac{22}{7}\times 1.75\times 10$
$=$ $2\times 22\times 0.25\times 10=110 m^{2}$
Then cost of plastering curved surface area $=$ $110\times 40=$ Rs. $4400$.

The height of a hollow cylinder is $7 cm$ and its radius is $3.5 cm$. Then the surface area is

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

  1. $231{ cm }^{ 2 }$

  2. $154{ cm }^{ 2 }$

  3. $308{ cm }^{ 2 }$

  4. $115.5{ cm }^{ 2 }$

Show answer
Correct Option: A
Explanation:

Given : radius $r=3.5 cm$ and height $h=7cm$

Surface area of a hollow cylinder $=2\pi r(h+r)$
                                                        $=2\times 3.14\times 3.5(7+3.5)$
                                                        $=230.79cm^2\approx 231$
$\therefore$ Surface area $=231cm^2$.

A magnet is in the form of a ring with inner diameter $4cm$ and outer diameter $6cm$. If the thickness of the magnet is $2cm$ . What is the cost of fabricating the surface of the magnet if the cost of fabrication per ${cm}^{2}$ is $Rs.10$

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

  1. $Rs.950$

  2. $Rs.945$

  3. $Rs.942$

  4. $Rs.1000$

Show answer
Correct Option: C

A flower pot is in the form of a hollow cylinder with a closed base with inner radius $2cm$ and outer radius $4cm$ . The height of the flower pot is $10cm$ . If the pot has to be polished find the cost of polishing if the cost of polishing per ${cm}^{2}$ is $Rs.2$.

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

  1. $Rs.276.32$

  2. $Rs.275$

  3. $Rs.270$

  4. $Rs.278.64$

Show answer
Correct Option: A
Explanation:

We have to first find the total surface area of the flower pot.
Outer radius $=4cm$
Inner radius $=2cm$
Height of flower pot $=10cm$
Total surface area $=A=2\pi Rh+2\pi rh+\pi { { R }^{ 2 } }+\pi { { r }^{ 2 } }$
$A=2\pi (R+r)h+\pi ({ R }^{ 2 }+{ r }^{ 2 })\ A=2\pi (4+2)2+\pi (16+4)\ A=24\pi +20\pi =44\pi \ A=40\pi =44\times 3.14=138.16{ cm }^{ 2 }$
Cost of fabrication per ${cm}^{2}$ $=Rs.2$
Total cost of fabrication $==138.16\times 2=Rs.276.32$

What will be the Inner surface area of a spherical shell of inner radius $15\ cm$ and outer radius $16\ cm$? (Correct upto 2 decimal places)

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

  1. $706.86\ {cm^2}$

  2. $804.25\ {cm^2}$

  3. $2827.43\ {cm^2}$

  4. $3216.99\ {cm^2}$

Show answer
Correct Option: C

The surface area of a solid sphere is always greater than the surface area of a hemisphere for the same value of radius.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
Let radius of solid sphere=radius of hemisphere$=r$
Then, S.A of solid sphere$=4\pi { r }^{ 2 }$
S.A of hemisphere$=2\pi { r }^{ 2 }$
$\therefore $S.A. of solid sphere$>$ S.A of hemisphere
Hence statement is true

A hemispherical bowl has inner radius $5cm$ and outer radius $6cm$. What will be the volume of solid enclosed between the two hemispheres? (Correct upto 2 decimal places)

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

  1. $904.78 \ {cm}^{3}$

  2. $523.60 \ {cm}^{3}$

  3. $381.18 \ {cm}^{3}$

  4. $190.59 \ {cm}^{3}$

Show answer
Correct Option: D
Explanation:

The volume of hemispherical shell$= \dfrac{2}{3}\pi*R^{3}-\dfrac{2}{3}\pi*r^{3}$
where R and r are the outer and inner radius of the hemisphere
On solving the equation we get Volume$= 190.59 \ {cm}^{3}$

The surface area of a solid spherical ball of diameter $10\ cm$ is equal to :

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

  1. $25\pi\ {cm}^2$

  2. $50\pi\ {cm}^2$

  3. $100\pi\ {cm}^2$

  4. $200\pi\ {cm}^2$

Show answer
Correct Option: C
Explanation:

Given: Diameter of the sphere $= 10\ cm$
Hence, Radius ($r$) of the sphere will be $5\ cm$

We know that, 
Surface area of the sphere is $4\pi r^2$
Therefore, Area will be $4\pi (5)^2 = 100\pi\ {cm}^2$

What will be the Inner and Outer radius of a spherical shell of inner surface area $452.39\ {cm}^2$ and outer surface area $804.25\ {cm}^2$ ? (Surface areas are accurate upto 2 decimal places)

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

  1. $6 \ cm, 7 \ cm$

  2. $7 \ cm, 8 \ cm$

  3. $6 \ cm, 8 \ cm$

  4. $7 \ cm, 9\ cm$

Show answer
Correct Option: C
Explanation:
Inner surface area$=4\pi { \left( inner\quad radius \right)  }^{ 2 }$
$\Rightarrow 4\pi { \left( { r } _{ 1 } \right)  }^{ 2 }=452.39cm^{2}\Rightarrow { r } _{ 1 }^{ 2 }=\cfrac { 452.39\times 7 }{ 4\times 22 } =35.99$
$\Rightarrow { r } _{ 2 }=\sqrt { 35.99 } =5.99\approx 6cm$
Outer surface area$=4\pi { \left( outer\quad radius \right)  }^{ 2 }$
$\Rightarrow 4\pi { \left( { r } _{ 2 } \right)  }^{ 2 }=804.25㎠\Rightarrow { r } _{ 2 }^{ 2 }=\cfrac { 804.25\times 7 }{ 4\times 22 } =63.97$
$\Rightarrow { r } _{ 2 }=\sqrt { 63.97 } =7.99\approx 8cm$

The value of radius for which the numerical value of total surface area of a sphere and the volume of sphere are equal, will be:(Consider the units of volume and surface area as ${cm}^3\  \text{and}\ {cm}^2$)

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

  1. $1cm$

  2. $2cm$

  3. $3cm$

  4. $4cm$

Show answer
Correct Option: C
Explanation:
Let radius of sphere be $'r'㎝$, then
TSA of sphere=volume of sphere
$\Rightarrow 4\pi { r }^{ 2 }=\cfrac { 4 }{ 3 } \pi { r }^{ 3 }\Rightarrow r=3cm$

The increase in the total surface area of a sphere of Radius ${R}$ when it is cut to make two hemispheres of same Radius will be equal to:

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

  1. $5\ \pi{R}^2$

  2. $4\ \pi{R}^2$

  3. $3\ \pi{R}^2$

  4. $2\ \pi{R}^2$

Show answer
Correct Option: D
Explanation:
Total surface area of sphere$=4\pi { R }^{ 2 }$
TSA of hemisphere=CSA of hemisphere+CSA of circle
$=2\pi { R }^{ 2 }+\pi { R }^{ 2 }=3\pi { R }^{ 2 }$
$\therefore $TSA of two hemisphere$=2\times 3\pi { R }^{ 2 }=6\pi { R }^{ 2 }$
Therefore, increase in TSA$=6\pi { R }^{ 2 }-4\pi { R }^{ 2 }=2\pi { R }^{ 2 }$

The height of a cylinder is  $14 { cm }$  and its  $ { CSA }$  is  $264 { cm } ^ { 2 },$  then cylinder is.........${ cm }^{ { 3 } }$

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

  1. $183$

  2. $396$

  3. $896$

  4. $968$

Show answer
Correct Option: A
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