A regular hexagon is inscribed in a circle. What is the ratio of the area of the circle to that of its portion not covered by the hexagon?

 \(\frac{2\pi }{2\pi  - 3\sqrt { 3}}\)

Area of circle = \(\pi \)r²

Not covered by the hexagon = area of circle - area of hexagon

                                         =  \(\pi \)r² - 6\(\frac{\sqrt { 3}}{4}\)a² 

Ratio = \(\frac{Area\; of\; circle}{Not\; covered\; area}\) = \(\frac{\pi r²}{\pi r² - 6\frac{\sqrt { 3}}{4}r²}\) = \(\frac{2\pi }{2\pi  - 3\sqrt { 3}}\)