We describe a quantum algorithm that can compute the electromagnetic scattering cross section of an arbitrary target. The algorithm solves the scattering problem using the finite element method. We show how one can prepare the initial quantum states, perform a quantum linear system solver subroutine, and measure the outcome in a manner that allows for efficient estimation of the cross section. To demonstrate the algorithm's functionality, we explicitly derive the classical oracles necessary to implement the quantum subroutines. This quantum algorithm can provide exponential speedup over the best classical algorithm, greatly improving the runtime and allowing for the modeling of far more complex objects than possible on a classical computer.