An analytic solution of the Förster energy transfer problem in two dimensions is presented for the case in which the orientation factor is independent of the donor-acceptor distance, and both the donors and acceptors are randomly distributed in a plane. A general solution based on the method of Förster is possible since all distances are measured in units of R0. The analytic solution is extended to the cases of donors embedded in structures that exclude acceptors, and donors that bind acceptors. The validity of the analytic solutions is demonstrated by comparison with numerical simulation calculations. Numerical approximations to the exact solutions are given for ease of computation. Specific applications to the case of fluorescence quenching of a membrane-bound donor by membrane-bound acceptors are presented.