Polygonal elements provide more flexibility in biased mesh generation of complex domains with a lesser number of elements in comparison to available unstructured triangular and quadrilateral elements. Furthermore, polygonal elements can provide more accurate solutions with larger element sizes. Polygonal elements are also used as transition elements to connect two dissimilar elements and to perform coarsening and refinement of structured meshes. In polygonal elements, the Wachspress shape functions based on rational polynomial are comparatively easy to develop and fulfill the partition of unity and Kronecker delta properties of conventional Finite Element Method (FEM) shape functions. While, a large number of studies using linear Wachspress shape functions can be found in the literature for linear elasticity, elastic fracture mechanics, and elasto-plasticity, works in literature on higher-order Wachspress shape functions are scarce. The higher-order functions fulfill the C1 continuity in a domain and can reduce the error in solution with a lesser number of elements more effectively. Based on this understanding, we have tried to develop the quadratic Wachspress shape function in this work. The higher-order interpolation is achieved by inserting middle nodes on each polygonal edge and then applying the same procedure as constructing rational polynomials for the linear Wachspress shape functions. For numerical integration of the weak form, we have applied the Gaussian quadrature rule on the triangulated polygons. The efficiency of the developed higher-order polygonal elements has been tested using patch test and beam bending problem in elastic and elasto-plastic analysis. The solutions have been compared with the conventional quadrilateral elements for elastic plane stress analysis, which demonstrate the workability of the higher-order polygonal elements.