The paper deals with the boundary value problem for the nonlinearly deformable composite cylinder with different types of boundary conditions. The stresses and displacements on both boundaries of the cylinder are constant, so their boundary average values for any area are constant and equal to the initial values. It should be noticed, that the solution of the boundary value problem is obtained without using nonlocal hypotheses about the composite material volume smallness by the angle for which the effective characteristics are calculated. In addition, the assumption of the composite material element smallness in the radial direction with respect to the thickness of the cylinder is used. It is established, that there is no possibility to consider plane stress state and plane strain of the cylinder separately from each other. Both of these states should be studied for analysis of stress-strain state according to Voigt and Reuss hypotheses. It is also shown that the solution of the Lame problem for a cylinder, which is derived, based on Voigt and Reuss hypotheses, is self-sufficient. Formulas, which describe stress-strain state of the composite cylinder, are derived based on this approximation.