We propose an approximation method to answer point-to-point shortest path queries in undirected graphs, based on random sampling and Voronoi duals. We compute a simplification of the graph by selecting nodes independently at random with probability $p$. Edges are generated as the Voronoi dual of the original graph, using the selected nodes as Voronoi sites. This overlay graph allows for fast computation of approximate shortest paths for general, undirected graphs. The time--quality tradeoff decision can be made at query time. We provide bounds on the approximation ratio of the path lengths as well as experimental results.The theoretical worst-case approximation ratio is bounded by a logarithmic factor. Experiments show that our approximation method based on Voronoi duals has extremely fast preprocessing time and efficiently computes reasonably short paths.