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 second square is

For the first case where the square (side being a cm) is inscribed in a circle, the square's diagonal is equal to the circle's diameter (2r).

Hence, 2r = a$$\sqrt{\ 2}$$, Hence a= r$$\sqrt{\ 2}$$

For the second case, where the square (side is b cm) is inscribed in the semi circle with the same radius, the radius, half of the square side, and a complete side will form a right angle triangle (from the mid point on the base of the semi circle). 

Hence, $$r^2=\ \ \frac{\ r^2}{4}\ +\ b^2$$, or b= $$\ \frac{\ 2r}{\sqrt{\ 5}}$$

Hence the area of square with side a to the area of square with side b is 5:2