In my earlier paper [4], an eigen-decompositions of the Laplacian operator is given on a unit regular hexagon with periodic boundary conditions. Since an exact decomposition with Dirichlet boundary conditions has not been explored in terms of any elementary form.In this paper, we investigate an approximate eigen-decomposition. The function space,corresponding all eigenfunction, have been decomposed into four orthogonal subspaces.Estimations of the first eight smallest eigenvalues and related orthogonal functions are given. In particulary we obtain an approximate value of the smallest eigenvalue λ1 ～29/40 π2 = 7.1555, the absolute error is less than 0.0001.