Based on the Wachspress generalized barycentric coordinates, we propose and analyze a polygonal finite volume element method (PFVEM) for solving the anisotropic diffusion equation on convex polygonal meshes. In particular, the PFVEM reduces to the classical P1-FVEM on triangular meshes and it is not identical to the classical Q1-FVEM on quadrilateral meshes. A new proof is given to Proposition 8 in [Adv. Comput. Math. 37 (2012) 417–439], a result that is crucial to the derivation of the interpolation error estimates in both H1 and H2 norms. The original proof is based on a certain geometric assumption, which is shown not always true by a counterexample. Moreover, for the error analysis of the PFVEM, we prove the H2 error estimate of the Wachspress interpolation. Under the coercivity assumption, the optimal H1 error estimate for the finite volume element solution is obtained. Several numerical examples are presented to show the efficiency and robustness of the proposed method for the heterogeneous and anisotropic diffusion problems on polygonal meshes.