Least-square approximation by polynomials, with respect to area measure on the regular hexagon, is useful in the construction and analysis of hexagonal optical elements. Notably the Keck Ten-Meter telescope will utilizes the main mirror composed of thirty-six hexagonal segments. By use of the representation theory of the symmetry group of the hexagon a method for constructing a family of orthogonal polynomials is derived. The computations can be carried out in rational (exact) arithmetic, and an appendix lists the polynomials and orthogonal expansions of monomials up to degree six.