Originally Posted by Richard

OK, it is my turn to pose a problem --

A regular polygon having N sides is circumscribed on a circle having unit radius. [Note: all sides of a regular polygon are equal, and all of its angles are equal. 'Circumscribed' means that the midpoints of each of the polygon's sides are tangent to the circle.]

As a function of N, find an expression for the fraction of the polygon's area that lies outside the circle. In other words, what fraction of the polygon would have to be removed to leave the circle?

Added challenge (ala Monsieurcoffee's previous problem): show a derivation of your expression that does not make use of transcendental functions such as sine, cosine, tangent, etc.