The Waldschmidt constant, introduced by Waldschmidt in 1975, has become an important invariant in commutative algebra and algebraic geometry. Much of the existing work has focused on establishing bounds and studying its asymptotic behavior. Nevertheless, explicit computations of Waldschmidt constants remain diffcult, even for relatively simple configurations of points in the projective plane. In this paper, we determine the Waldschmidt constant of the ideal defining three fat coordinate points in P². Our approach is based on describing the symbolic polyhedron of the ideal and expressing the computation of its Waldschmidt constant as a linear programming problem over this polyhedron.