A square is inscribed in a circle and another in a semi-circle of same radius. The ratio of the area of the first square to the area of the second square is :

Answer: Option B

Let the radius of each of the circle and the semi-circle be r units
Diagonal of the first square = 2r units
Let the side of the second be a units

Then,
$$\begin{array}{rl}& r^2=a^2+{\left(\frac{a}{2}\right)}^2\\ & \Rightarrow r^2=\frac{5a^2}{4}\\ & \Rightarrow a^2=\frac{4r^2}{5}\end{array}$$
∴ Ratio of the areas of the two squares :
$$\begin{array}{rl}& =\frac{\frac{1}{2}\times {\left(2r\right)}^2}{a^2}\\ & =\frac{2r^2}{\left(\frac{4r^2}{5}\right)}=\frac{5}{2}\\ & =\frac{5}{2}\\ & =5:2\end{array}$$