This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L-2-norm and the H-1-seminorm. 2022 Elsevier B.V. All rights reserved.