We analyze a boundary-value problem of the linear theory of elasticity for a plane stressed state of a curvilinear unequal-sided trapezoid with an arbitrary geometry of sides. We approximate the system of equations of the plane problem of the theory of elasticity and corresponding boundary conditions by a finite system of one-dimensional boundary-value problems. In order to do this, we use a process of reduction of two-dimensional relationships to one-dimensional ones based on the expansion of unknown functions into the Fourier-Legendre series and minimization of the corresponding Reissner functional. The convergence of the approximate solution of nth order to known exact solution is proven by an example of the Filon problem for a rectangular plate. A numerical solution of the equations of nth order of approximation for orthotropic curvilinear unequal-sided trapezoid is derived.