The limit of such a regular n-gon is a circle with radius pi.

In polar coordinates if can be plotted as

r = 2*pi*cos(theta)

I'm tempted to say that the sought mean length would be

Integral{0 to pi}(2*pi*cos(theta) d(theta) / pi

but this does not take into consideration any clustering of the angles involved; it assumes the angles the diagonals make with tangents at a given point are uniformly distributed.  So I don't think it's worthwhile to use this integral.