A square and a regular hexagon are drawn such that all the vertices of the square and the hexagon are on a circle of radius r cm. The ratio of area of the square and the hexagon is

$$\eqalign{ & {\text{Diagonal of square}} = \sqrt 2 a = 2r \cr & \therefore a = \sqrt 2 r \cr & {\text{Area of square}} = {a^2} = {\left( {\sqrt 2 r} \right)^2} = 2{r^2} \cr} $$

$$\eqalign{ & {\text{Side of hexagon}} = a = r \cr & {\text{Area of hexagon}} = 6\frac{{\sqrt 3 }}{4}{a^2} = 3\frac{{\sqrt 3 }}{2} \times {r^2} \cr & {\text{Required ratio}} = 2{r^2}:3\frac{{\sqrt 3 }}{2}{r^2} = 4:3\sqrt 3 \cr} $$