  We consider consecutive random subdivision of polygons described as follows.
Given an initial convex polygon with $d\ge 3$ edges, we choose a point at
random on each edge, such that the proportions in which these points divide
edges are i.i.d. copies of some random variable $\xi$. These new points form a
new (smaller) polygon. By repeatedly implementing this procedure we obtain a
sequence of random polygons. The aim of this paper is to show that under very
mild non-degenerateness conditions on $\xi$, the shapes of these polygons
eventually become "flat" The convergence rate to flatness is also investigated;
in particular, in the case of triangles ($d=3$), we show how to calculate the
exact value of the rate of convergence, connected to Lyapunov exponents. Using
the theory of products of random matrices our paper greatly generalizes the
results of Volkov (2013) which are achieved mostly by using ad hoc methods.
