We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing over-approximation tools tend to be conservative or computationally expensive. In this work, we exactly characterize the convex hulls of reachable sets as the convex hulls of solutions of an ordinary differential equation from all possible initial values of the disturbances. This finite-dimensional characterization unlocks a fast sampling-based method to accurately overapproximate reachable sets. We give applications to neural feedback loop analysis and robust model predictive control.