Network data, commonly used throughout the physical, social, and biological sciences, consist of nodes (individuals) and the edges (interactions) between them. One way to represent network data's complex, high-dimensional structure is to embed the graph into a low-dimensional geometric space. The curvature of this space, in particular, provides insights about the structure in the graph, such as the propensity to form triangles or present tree-like structures. We derive an estimating function for curvature based on triangle side lengths and the length of the midpoint of a side to the opposing corner. We construct an estimator where the only input is a distance matrix and also establish asymptotic normality. We next introduce a novel latent distance matrix estimator for networks and an efficient algorithm to compute the estimate via solving iterative quadratic programs. We apply this method to the Los Alamos National Laboratory Unified Network and Host dataset and show how curvature estimates can be used to detect a red-team attack faster than naive methods, as well as discover non-constant latent curvature in co-authorship networks in physics. The code for this paper is available at https://github.com/SteveJWR/netcurve, and the methods are implemented in the R package https://github.com/SteveJWR/lolaR.