The condition of incompressibility for an isotropic linearly elastic material seriously limits the application of the classical hypotheses of the theory of bending of plates, formulated for small strains and displacements. It is assumed that such a strong kinematic condition as a condition of invariability of the volume must certainly be met. When bending an incompressible round plate by a force load, it is shown that the application of certain Kirchhoff hypotheses is associated with the boundary conditions of the problem: for some conditions it is possible to use separate hypotheses, for others it is necessary to completely abandon, and to obtain relatively simple solutions to build models of calculation using other hypotheses that are not inconsistent with the condition of invariability of the volume. So, for example, at rigidly fixed of a plate on two contours (or on one for a continuous plate) absence of deformation in the transverse direction in relation to bases should be considered not as a hypothesis, and a consequence of an incompressibility condition. Hypotheses of no shift in the plane and normal stress are not used because of their incompatibility with the incompressibility condition. The temperature load has its own characteristics, which is why the temperature problem requires a non-standard approach in its solution. Note, first of all, that the invariability of the volume is kept for the elastic component of the total deformation, but to the temperature component has nothing in relation. Therefore, the total deformation of the volume change is not zero, since it is related to the temperature field. Therefore, this connection is not a condition of incompressibility, but it is an important kinematic condition for the incompressible plate. The principle of using different hypotheses under different boundary conditions is kept here. At the same time, the absence of transverse deformation is a hypothesis even under the condition of rigidly fixed of the contours due to the variability of the volume of the plate under bending temperature loading.