Quoting AA (Noname) How did you get the identities you've written?
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.