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 :

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