Accurate animal cell division requires precise coordination of changes in the structure of the microtubule-based spindle and the actin-based cell cortex. Here, we use a series of perturbation experiments to dissect the relative roles of actin, cortical mechanics, and cell shape in spindle formation. We find that, whereas the actin cortex is largely dispensable for rounding and timely mitotic progression in isolated cells, it is needed to drive rounding to enable unperturbed spindle morphogenesis under conditions of confinement. Using different methods to limit mitotic cell height, we show that a failure to round up causes defects in spindle assembly, pole splitting, and a delay in mitotic progression. These defects can be rescued by increasing microtubule lengths and therefore appear to be a direct consequence of the limited reach of mitotic centrosome-nucleated microtubules. These findings help to explain why most animal cells round up as they enter mitosis.