Many plant species have thin leaf blades and axisymmetric elongating organs, such as stems and roots. Here, we provide evidence that a single mechanism can account for these fundamentally different organ shapes. We show that the three-dimensional geometry of leaves is governed by microtubule alignment along predicted mechanical stress patterns along internal walls. Depending on the primary shape of the primordium, this process has the potential to amplify an initial degree of flatness, or to promote the formation of nearly axisymmetric, mostly elongating organs. This mechanism may explain leaf evolution from branches, which is alternative to Zimmermann’s influential, but widely questioned, telome theory.