  We study the probability distribution of the area and the number of vertices
of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of
our approach is that it yields uniform estimates for all convex sets
$K\subset\mathbb{R}^2$ without imposing any regularity conditions on the
boundary $\partial K$. Our main result is a central limit theorem for both the
area and the number of vertices, settling a well-known conjecture in the field.
We also obtain asymptotic results relating the growth of the expectation and
variance of these two functionals.
