We have used atomic force microscopy to study in detail the shape of InAs quantum dots (QDs) grown on GaAs (001) substrates. The QDs are fabricated by partial capping and a subsequent annealing process at low temperature. We observe circular, non-elongated QDs, with rim structures on top. The majority of the QDs located on a step edge have an anisotropic rim structure with a crescent shape, whereas approximately 50% of QDs located on terraces have isotropic rim structures with a ring shape. These results indicate that the rims of non-elongated QDs are not always isotropic, and the formation of an isotropic or anisotropic rim structure is dependent on the location of the QD.