Tag: 2d and 3d figures

Questions Related to 2d and 3d figures

A square sheet of paper is converted into a cylinder by rolling it along its length. What is the ratio of the base radius to side of the square ?

maths area of complex plane figures 2d and 3d figures

  1. $\displaystyle \frac{1}{2\pi}$

  2. $\displaystyle \frac{\sqrt{2}}{\pi}$

  3. $\displaystyle \frac{1}{\sqrt{2\pi }}$

  4. $\displaystyle \frac{1}{\pi}$

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Correct Option: A
Explanation:

Let length of each side of the square $=a$

Base radius of cylinder $=r$
Surface area of sheet $={ a }^{ 2 }$
Surface area of cylinder $=2\pi rh$
But height $h=a$, because the square sheet is rolled it along its length.
Thus, surface area of cylinder $=2\pi ra$
Therefore, ${ a }^{ 2 }=2\pi ra\Rightarrow a=2\pi r\Rightarrow \dfrac { r }{ a } =\dfrac { 1 }{ 2\pi  } $

A polygon has 44 diagonals, The number of its sides is

maths area of complex plane figures 2d and 3d figures

  1. 11

  2. 10

  3. 8

  4. 7

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Correct Option: A
Explanation:
Number of diagonals in a polygon $=\cfrac{n(n-3)}2$
$\implies 44=\cfrac{n(n-3)}2$
$\implies n^2-3n-88=0$
$\implies (n-11)(n-8)=0$
$\implies n=11$ or $n=-8$
Therefore, number of sides in a polygon $=11.$
Hence, A is the correct option.

Four circular cardboard pieces of radii $7 cm$ are placed on a paper in such a way that each piece touches other two pieces. The area of the  region enclosed between these pieces   is

maths area of complex plane figures 2d and 3d figures

  1. $42$ $cm^2$

  2. $21$  $cm^2$

  3. $84$ $cm^2$

  4. $96$ $cm^2$

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Correct Option: A
Explanation:

The diameter of circle =$2\times 7=14$ cm

The circles together formed a shape  square diameter of 2 circle to get from a side =$2\times 14=28$ cm 
Then area of square =$a^{2}=(28)^{2}=784 cm^{2}$
And area of each  circle =$\pi r^{2}=\frac{22}{7}\times (7)^{2}=154 cm^{2}$
So area of four circles =$4\times 154=616 cm^{2}$
Then area of region enclosed between these pieces $= 784-616=168$ sq cm
Then area of region enclosed between one  pieces=$\frac{168}{4}=42 cm^{2}$

The ratio of the slant height of two right cones of equal base is 3 : 2 then the ratio of their volumes is

maths area of complex plane figures 2d and 3d figures

  1. $1:4$

  2. $9:4$

  3. $3:2$

  4. None

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Correct Option: B
Explanation:

Let the radius of both the cones be r, slant height of first cone $ = 3a $; slant height of second cone $ = 2a $

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

Hence, for cone 1,  $ 3a = \sqrt { { h 1 }^{ 2 }+  {r}^{ 2 } } $ 

$ 9{a}^{2} = { h 1 }^{ 2 } + {r}^{ 2 }$

$ { h 1 }^{ 2 } = 9{a}^{2} - {r}^{ 2 } $

$ h 1 = \sqrt {9{a}^{2} - {r}^{ 2 }} $

For cone 1,  $ 2a = \sqrt { { h 2 }^{ 2 }+  {r}^{ 2 } } $ 

$ 4{a}^{2} = { h 2 }^{ 2 } + {r}^{ 2 }$

$ { h 2 }^{ 2 } =4{a}^{2} - {r}^{ 2 } $

$ h 2 = \sqrt {4{a}^{2} - {r}^{ 2 }} $
Now, ratio of their volumes is calculated as:
$V _{ 1 } : V _{ 2 }$
$ \frac { 1 }{ 3 }\pi {r }^{ 2 }h 1 = \frac { 1 }{ 3 } \pi { r }^{2 }h 2 $
$ h 1 : h 2 $
$ \sqrt {9{a}^{2} - {r}^{ 2 }} : \sqrt {4{a}^{2} - {r}^{ 2 }}  $
$9{a}^{2} - {r}^{ 2 }: 4{a}^{2} - {r}^{ 2 } $
$ 9{a}^{2} : 4{a}^{2} $
$ 9:4 $