In this paper, we propose a new algorithm that computes the radius and the diameter of a weakly connected digraph G=(V,E), by finding bounds through heuristics and improving them until they are validated. Although the worst-case running time is O(|V||E|), we will experimentally show that it performs much better in the case of real-world networks, finding the radius and diameter values after 10–100 BFSs instead of |V| BFSs (independently of the value of |V|), and thus having running time O(|E|) in practice. As far as we know, this is the first algorithm able to compute the diameter of weakly connected digraphs, apart from the naive algorithm, which runs in time Ω(|V||E|) performing a BFS from each node. In the particular cases of strongly connected directed or connected undirected graphs, we will compare our algorithm with known approaches by performing experiments on a dataset composed by several real-world networks of different kinds. These experiments will show that, despite its generality, the new algorithm outperforms all previous methods, both in the radius and in the diameter computation, both in the directed and in the undirected case, both in average running time and in robustness. Finally, as an application example, we will use the new algorithm to determine the solvability over time of the “Six Degrees of Kevin Bacon” game, and of the “Six Degrees of Wikipedia” game. As a consequence, we will compute for the first time the exact value of the radius and the diameter of the whole Wikipedia digraph.