Stats: Find the shaded area

Okay, here's the old stats challenge. Imagine a regular (all equal length sides) polygon inscribed in a circle. By inscribed, we mean that all of the vertices of the polygon lie on the circle. It's not too hard to find the ratio of the polygon's area to the circle's area for some of the simpler shapes, especially a square and an octagon. I was also able to do a triangle and hexagon with some additional tricks (remember the 30-60-90 triangle?). The ratio of the polygon's area to the circle's area are as follows for these shapes:

Triangle: 3/(2*pi)
Square: 2/pi
Hexagon: 3*sqrt(3)/(2*pi)
Octagon: 2*sqrt(2)/pi

The Challenge:
For either a regular pentagon or heptagon (that's 7 sides) inscribed in a circle, find the ratio of the polygon's area to the area of the circle. For an extra bonus, find the pattern relating a general n-sided polygon's area to a circle in which it is inscribed. Note that I expect exact answers, in the forms above with square roots, pi's, and numbers in fractions, rather than simplified decimals.

This is a trigonometric problem more than a stats one, but it's important because a similar problem, the ratio of the perimeters of these polygons to the radius of the circle, formed the basis of ancient attempts to find the value of pi!