We present a new algorithm, trimed, for obtaining the medoid of a set, that is the element of the set which minimises the mean distance to all other elements. The algorithm is shown to have, under weak assumptions, complexity O(N^(3/2)) in R^d where N is the set size, making it the first sub-quadratic exact medoid algorithm for d>1. Experiments show that it performs very well on spatial network data, frequently requiring two orders of magnitude fewer distances than state-of-the-art approximate algorithms. We show how trimed can be used as a component in an accelerated K-medoids algorithm, and how it can be relaxed to obtain further computational gains with an only minor loss in quality.