Leaves vary from planar sheets and needle-like structures to elaborate cup-shaped traps. Here, we show that in the carnivorous plant Utricularia gibba, the upper leaf (adaxial) domain is restricted to a small region of the primordium that gives rise to the trap's inner layer. This restriction is necessary for trap formation, because ectopic adaxial activity at early stages gives radialized leaves and no traps. We present a model that accounts for the formation of both planar and nonplanar leaves through adaxial-abaxial domains of gene activity establishing a polarity field that orients growth. In combination with an orthogonal proximodistal polarity field, this system can generate diverse leaf forms and account for the multiple evolutionary origins of cup-shaped leaves through simple shifts in gene expression.