We determine the equilibrium morphology of a strained solid film for the case where it wets the substrate (Stranski-Krastanow growth). Using a continuum elasticity model with isotropic surface energy and equal elastic constants in the film and substrate, we determine an asymptotic solution for the axisymmetric three-dimensional equilibrium shape of a small island, where the height is much less than the width, resulting in a codimension-two free boundary problem. This codimension-two free boundary problem can be reformulated as an integro-differential equation in which the island width appears as an eigenvalue. The solutions to the resulting integro-differential eigenvalue problem consist of a discrete spectrum of island widths and associated morphological modes, which are determined using a rapidly converging Bessel series. The lowest-order mode is energetically preferred and corresponds to the quantum dot morphology. Our predictions of quantum dot width compare favorably with experimental data in the GeSi/Si system. The higher-order modes, while not minimum-energy configurations, are similar to 'quantum ring` and 'quantum molecule` morphologies observed during the growth of strained films.