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

Answer: Option C

Solution

Answer: Option C

Solution:

$$\begin{array}{rl}& \text{Diagonal of square}=\sqrt{2}a=2r\\ & \therefore a=\sqrt{2}r\\ & \text{Area of square}=a^2={\left(\sqrt{2}r\right)}^2=2r^2\end{array}$$

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