Question

The perimeter of a circle is equal to twice that of square, then the ratio of their areas is: -
a) 22/7
b) 14/11
c) 7/22
d) 56/11​

Answer 1

Given : The perimeter of a circle is equal to twice that of square,  

To Find :     ratio of their areas is: -

Solution:

Radius of circle  =  r

Side of square = a

Perimeter of circle =2πr

Perimeter of square = 4a

2πr = 2 * 4a

=> r = 4a/π

Area of circle  = πr²  = π(4a/π)²  = 16 a²/π

Area of  square = a²

Ratio  =  16 /π

16/(22/7)

= 7 * 16 /22

= 56/11

ratio of their areas is: -  56/11​

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Answer 2

Given,

The perimeter of a circle is equal to twice the square.

Solution,

Assume that the radius of the circle is$\[r\]$ and side of the square is $\[a\].$

Therefore,

$\[\begin{array}{l} \Rightarrow 2 \times \frac{{22}}{7} \times r = 2 \times 4 \times a\\ \Rightarrow \frac{r}{a} = \frac{{2 \times 4 \times 7}}{{2 \times 22}}\\ \Rightarrow \frac{r}{a} = \frac{{14}}{{11}}\end{array}\]$

Now, the ratio of their area is

$\[\begin{array}{l} \Rightarrow \frac{{\pi {r^2}}}{{{a^2}}}\\ \Rightarrow \pi \times {\left( {\frac{r}{a}} \right)^2}\end{array}\]$

Apply values.

$\[\begin{array}{l} \Rightarrow \frac{{22}}{7} \times {\left( {\frac{{14}}{{11}}} \right)^2}\\ \Rightarrow \frac{{56}}{{11}}\end{array}\]$

Hence, the correct option is (d) i.e. $\[\frac{{56}}{{11}}\].$