Colloidal semiconductor nanocrystals are commonly grown with a shell of a second semiconductor material to obtain desired physical properties, such as increased photoluminescence quantum yield. However, the growth of a lattice-mismatched shell results in strain within the nanocrystal, and this strain has the potential to produce crystalline defects. Here, we study CdSe/CdS core/shell nanorods as a model system to investigate the influence of core size and shape on the formation of stacking faults in the nanocrystal. Using a combination of high-angle annular dark-field scanning transmission electron microscopy and pair-distribution-function analysis of synchrotron X-ray scattering, we show that growth of the CdS shell on smaller, spherical CdSe cores results in relatively small strain and few stacking faults. By contrast, growth of the shell on larger, prolate spheroidal cores leads to significant strain in the CdS lattice, resulting in a high density of stacking faults.