Computing the exact area of the union of an arbitrary number of circles is a challenging problem, since the union is generally non-convex and may be composed of multiple non-overlapping regions. In this paper, we propose tackling this problem by using graph-theoretical concepts and Green’s Theorem for exact area computation. Moreover, we show the implementation of this method for a rapid area coverage application with unmanned aerial vehicles (UAVs). Maximizing the area covered using multiple agents is difficult because fast solutions to large-scale optimization problems are sought. In our solution method, we present a hierarchical control framework. On the upper layer, a high-level controller performs centralised computation to determine the optimal UAV locations to maximize the area covered. On the bottom level, we adopt a decentralised approach by implementing multiple local controllers to tackle the trajectory planning and collision avoidance for each agent individually using Nonlinear Model Predictive Control (NMPC). Numerical experiments show that our method for computing the covered area can reduce the computational time required to solve the optimal positioning problem by more than two orders of magnitude when compared to a Monte-Carlo method. The trajectory planning problem was tested for up to 13 agents and the run-time was on the order of milliseconds, demonstrating the suitability for real-time implementation of the presented framework.