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Tubular epithelia are a basic building block of organs and a common site of cancer occurrence1–4. During tumorigenesis, transformed cells overproliferate and epithelial architecture is disrupted. However, the biophysical parameters that underlie the adoption of abnormal tumour tissue shapes are unknown. Here we show in the pancreas of mice that the morphology of epithelial tumours is determined by the interplay of cytoskeletal changes in transformed cells and the existing tubular geometry. To analyse the morphological changes in tissue architecture during the initiation of cancer, we developed a three-dimensional whole-organ imaging technique that enables tissue analysis at single-cell resolution. Oncogenic transformation of pancreatic ducts led to two types of neoplastic growth: exophytic lesions that expanded outwards from the duct and endophytic lesions that grew inwards to the ductal lumen. Myosin activity was higher apically than basally in wild-type cells, but upon transformation this gradient was lost in both lesion types. Three-dimensional vertex model simulations and a continuum theory of epithelial mechanics, which incorporate the cytoskeletal changes observed in transformed cells, indicated that the diameter of the source epithelium instructs the morphology of growing tumours. Three-dimensional imaging revealed that—consistent with theory predictions—small pancreatic ducts produced exophytic growth, whereas large ducts deformed endophytically. Similar patterns of lesion growth were observed in tubular epithelia of the liver and lung; this finding identifies tension imbalance and tissue curvature as fundamental determinants of epithelial tumorigenesis. Three-dimensional imaging of mouse pancreatic ducts before and after oncogenic transformation reveals that epithelial tumorigenesis is determined by the relationship between tissue curvature and apical–basal mechanical tension.