TITLE: What is the expected area of a polygon whose vertices lie on a circle?
QUESTION [13 upvotes]: I came across a nice problem that I would like to share.
Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly distributed.

REPLY [3 votes]: A simple asympotics, for large $N$: the central angles $x_i$ (which sum up to $2\pi$, and hence are not independent), can be approximated as iid exponentials $y_i$ with mean $\alpha=2 \pi /N$ (a similar procedure as "Poissonization" for discrete variables - also analogous to change ensemble in statistical physics).
Then, because the area of each triangle is $ \sin(x_i)/2$, we have that the expected area is
$$E(A) = \frac{1}{2}\sum_{i=1}^N E[\sin(x_i)] \approx \frac{N}{2} E[\sin(y_i)] = \frac{N}{2} \int_0^{\infty} \sin(y) \frac{1}{\alpha} \exp{\left(-\frac{y}{\alpha}\right)} \,dy = \frac{N}{2} \frac{\alpha}{1+\alpha^2}$$
So 
$$E(A)\approx  \frac{\pi}{1+(2 \pi/N)^2}$$