We study the problem of designing fault-tolerant routings for a communication network which is a triconnected planar network of processors in the surviving route graph model. The surviving route graph for a graph G, a routing ρ anda set of faults F is a directed graph consisting of nonfaulty nodes with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph couldb e one of the fault-tolerance measures for the graph G andt he routing ρ. In this paper, we show that we can construct a routing for any triconnectedpl anar graph with a triangle face such that the diameter of the surviving route graphs is two(thus optimal) for any faults F(|F| ≤ 2). We also show that the optimal routing can be computedin linear time.