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 :

Answer: Option B

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