Abstract The cost of obtaining solutions to problems governed by the Helmholtz equation in both interior and exterior domains by means of boundary element and finite element methods is studied and compared. The main emphasis is on the computational effort required to solve the systems of equations emanating from the two methods. Boundary element methods require fewer equations to be solved by virtue of the fact that only boundaries are discretized. These equations, however, are less structured than those of finite element methods and hence cost-effectiveness is not as clear cut as might be expected. Both direct and iterative solution techniques are examined. For interior problems finite element methods are more economical on most practical configurations. Finite elements also appear to possess a certain computational advantage on the exterior problems examined, and, in general, are definitely competitive with boundary element methods. The cost-effectiveness of the two solution strategies is examined. Some issues of equation formation are also addressed.