Across development, the formation of fluid-filled lumina enclosed by epithelial tissues is inextricably linked to the establishment of apicobasal cell polarity. Here, we reveal how the mechanical differences between cell surfaces resulting from this polarity enter the interplay of lumen pressure, mechanics of the cell cortex, and cell-cell adhesion that determines lumen morphology: Madin–Darby canine kidney cysts under tight junction perturbations display a shape instability of apical surfaces that is associated with changes of cyst pressure and apical belt tension [Mukenhirn et al., Dev. Cell59, 2886 (2026) 10.1016/j.devcel.2024.07.016]. Here, we discover that the cysts also respond to these perturbations by significant modulations of lateral and basal surface tensions. We develop a mean-field three-dimensional vertex model of these cysts that reproduces the experimental shape space and instability quantitatively. This model explains the observed increase of lateral contractility to be a cellular response that counters the instability. Our work thus shows how regulation of the mechanics of all cell surfaces conspires to control lumen morphology.