We present a new incremental algorithm for constructing the union of *n* triangles in the plane. In our experiments, the new algorithm, which we call the Disjoint-Cover (DC) algorithm, performs significantly better than the standard randomized incremental construction (RIC) of the union. Our algorithm is rather hard to analyze rigorously, but we provide an initial such analysis, which yields an upper bound on its performance that is expressed in terms of the expected cost of the RIC algorithm. Our approach and analysis generalize verbatim to the construction of the union of other objects in the plane, and, with slight modifications, to three dimensions. We present experiments with a software implementation of our algorithm using the CGAL library of geometric algorithms.