Say the side length of hexagon = 1, so perimeter = 6. The circumference of circle must be 6, so the radius is 2 * pi * r = 6 and r = 3/pi.

The area of the hexagon is 3sqrt(3)/2 by the formula (or draw 6 equliateral triangles). The areae of circle is 9/pi.

The ratio of hexagon to circle is (3sqrt(3)/2)/(9/pi) = (3sqrt(3) * pi) / (18) = (sqrt(3) * pi) / 6.

You are very welcome!

:P