  Fix a container polygon $P$ in the plane and consider the convex hull $P_n$
of $n\geq 3$ independent and uniformly distributed in $P$ random points. In the
focus of this paper is the vertex number of the random polygon $P_n$. The
precise variance expansion for the vertex number is determined up to the
constant-order term, a result which can be considered as a second-order
analogue of the classical expansion for the expectation of R\'enyi and Sulanke
(1963). Moreover, a sharp Berry-Esseen bound is derived for the vertex number
of the random polygon $P_n$, which is of the same order as the square-root of
the variance. The main idea behind the proof of both results is a decomposition
of the boundary of the random polygon $P_n$ into random convex chains and a
careful merging of the variance expansions and Berry-Esseen bounds for the
vertex numbers of the individual chains.
