OK, I got it. I'll write my explanation because it doesn't need any graphical aids.

Yes, but only if the height or the perimeter have a fixed value. I solved it using calculus and limits: The sum of angles is 180(n-2), therefore every angle in a regular polygon is 180(n-2)/n. As n approaches infinity, 180(n-2)/n approaches 180. The side is equal to P/n (P is the perimeter), and if P is a fixed number then as n approaches infinity the side approaches zero. How did you get the identities you've written? Anyway, your prize is a link on my homepage and six comments or three reviews (constructive criticism). This might interest you: http://www.youtube.com/watch?v=jG7vhMMXagQ

Which identities? the one for the heights? "I'm still trying to ask him for a longer explanation" I think 360/n, the top inside angle of the triangle with a side (one of the n) as it's base, lets say the other two side are of length "r". Now cut this triangle in half, with a line perpendicular to the base which goes through the vertex of the top of the triangle. It's easy to see from http://en.wikipedia.org/wiki/Trigonometric_functions that side_length/2 = r/csc(180/n) so side_length goes to 0 as n goes to infinity. Likewise 180 - 360/n goes 180.

:^) see http://mocpages.com/moc.php/92742 as the number of sides increases the length decreases, so the angle becomes 180 and the length goes to 0. It becomes a circle.

Are you interested in mathematics in general, or just geometry? Because I have an interesting mathematic puzzle: Suppose that you have a convex regular polygon with an infinite number of sides. What is the size of its angle and what is the size of its side?

This looks like it was fun to build.. has a very cool look to it.

@Thomas - not with C70 - that's a little more oblong. Maybe you could play rugby with it. C60 has the same shape and symmetry as a soccer ball.

Cool! can you play soccer with it?