Answer: Option C
$$\eqalign{
& 2\pi R = 100 \cr
& R = \frac{{100}}{{2\pi }} = \frac{{50}}{\pi } \cr
& R = \frac{1}{2} \times {\text{diagonal}} \cr
& \Rightarrow {\text{Diagonal}} = 2R \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2 \times 50}}{\pi } \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{100}}{\pi } \cr} $$
$$\eqalign{
& \therefore {\text{Area of the square}} = \frac{1}{2} \times {\left( {{\text{diagonal}}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2} \times {\left( {\frac{{100}}{\pi }} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{{\sqrt 2 }} \times \frac{{100}}{\pi } \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{50\sqrt 2 }}{\pi }cm \cr} $$