The ratio of the areas of the in-circle and the circum-circle of a square is :

Answer: Option A

Let r₁ and r₂ be the radii of the in-circle and circum-circle of a square respectively and let each side of the square be a.
Then,
$$\begin{array}{rl}& r_1=\frac{a}{2}\\ & r_2=\frac{1}{2}\times \text{diagonal of the sequence}\\ & r_2=\frac{1}{2}\times \sqrt{2}a\\ & r_2=\frac{\sqrt{2}a}{2}cm\end{array}$$
∴ Required ratio :
$$\begin{array}{rl}& =\frac{\pi \times {\left(\frac{a}{2}\right)}^2}{\pi \times {\left(\frac{\sqrt{2}a}{2}\right)}^2}\\ & =1:2\end{array}$$