  We establish three criteria of hyperbolicity of a graph in terms of ``average
width of geodesic bigons''. In particular we prove that if the ratio of the Van
Kampen area of a geodesic bigon $\beta$ and the length of $\beta$ in the Cayley
graph of a finitely presented group $G$ is bounded above then $G$ is
hyperbolic.
  We plan to use these results to characterize hyperbolic groups in terms of
random walks.
