The perimeter of a square and a regular hexagon are equal. The ratio of the area of the hexagon to the area of the square is :

Side of square = a units
Side of hexagon = b units
According to the question,
$$\eqalign{ & 4a = 6b \cr & \Rightarrow \frac{a}{b} = \frac{6}{4} = \frac{3}{2} \cr} $$
$$\eqalign{ & \therefore \frac{{{\text{Area of hexagon}}}}{{{\text{Area of square}}}} \cr & = \frac{{6 \times \frac{{\sqrt 3 }}{4} \times {b^2}}}{{{a^2}}} \cr & = \frac{{6 \times \sqrt 3 \times 2 \times 2}}{{4 \times 3 \times 3}} \cr & = \frac{{2\sqrt 3 }}{3} \cr} $$
Hence required ratio = $$2\sqrt 3 :3$$