Abstract We give asymptotically sharp upper bounds on the radius and diameter of (i) a connected graph, (ii) a connected triangle-free graph, (iii) a connected C4-free graph of given order, minimum degree, and maximum degree. We also give better bounds on the radius and diameter for triangle-free graphs with a given order, minimum degree and a given number of distinct terms in the degree sequence of the graph. Our results improve on old classical theorems by Erd˝os, Pach, Pollack and Tuza [Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47 (1989), 73-79] on radius, diameter and minimum degree.