A K-surface is a surface whose Gauss curvature K is equal to a positive constant. In this paper, we will consider K-surfaces that are defined by a nonlinear boundary value problem. In this setting, existence follows from some recent results on nonlinear second-order elliptic partial differential equations. The analytical techniques used to establish these results motivate effective numerical methods for computing K-surfaces. In theory, the solvability of the boundary value problem reduces to the existence of a subsolution. In an analogous way, if an approximate numerical subsolution can be determined, then the corresponding K-surface can be computed. We will consider two boundary value problems. In the first problem, the K-surface is a graph over a plane. In the second problem, the K-surface is a radial graph over a sphere. From certain geometrical considerations, it follows that there is a maximum allowable Gauss curvature Kmax for these problems. The principal results in this paper a...