The paper considers a physically linear mathematical model of an isotropic circular plate-membrane with a non-deformable central disk, concentrated load and zero bending stiffness with the account for finite displacements. On this basis, the extreme problem of determining the rational geometric parameters of an elastic element from the condition of the target sensitivity function maximum with the equation of constraint in the form of the Huber- Hencky-Mises strength energy hypothesis is solved. The analytical study of the influence of the Poisson’s ratio on the basic optimal dimensionless characteristic of the membrane, which is the ratio of radii, in comparison with the known calculation by the formulas of the classical linear theory of transverse bending of rigid plates is presented. The results of the work can be used in the process of design of high-precision capacitive, inductive and strain gauges of membrane type, widely used in mechanical engineering, aviation, instrument engineering and construction when designing pressure tanks with controlled overpressure of gas or liquid.