The flash method is the standard technique to measure the thermal diffusivity of solid samples. It consists of heating the front surface of an opaque sample by a brief light pulse and detecting the temperature evolution at its rear surface. The thermal diffusivity is obtained by measuring the time corresponding to the half maximum of the temperature rise, which only depends on the sample thickness and thermal diffusivity through a simple formula. Up to now, the flash method has been restricted to flat samples. In this work, we extend the flash method to measure the thermal diffusivity of nonflat samples. In particular, we focus on plates with cylindrical and spherical shapes. The theoretical model indicates that the same expression for flat samples can also be applied to cylindrical and spherical plates, except for extremely curved samples. Accordingly, a curvature limit for the application of the expression for flat samples is established. Flash measurements on lead foils of cylindrical shape confirm the validity of the model.