  Global physical properties of random media change qualitatively at a
percolation threshold, where isolated clusters merge to form one infinite
connected component. The precise knowledge of percolation thresholds is thus of
paramount importance. For two dimensional lattice graphs, we use the universal
scaling form of the cluster size distributions to derive a relation between the
mean Euler characteristic of the critical percolation patterns and the
threshold density $p_c$. From this relation, we deduce a simple rule to
estimate $p_c$, which is remarkably accurate. We present some evidence that
similar relations might hold for continuum percolation and percolation in
higher dimensions.
