The two common critical probabilities for a lattice graph L are the cluster size critical probability pH (L) and the mean cluster size critical probability pT (L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH (H) = pT (H) = 1–2 sin (π/18) and PH (T) = pT (T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.