This paper deals with the development of numerical methods for determining the strongest cantilever beam having a constant volume of the beam material. An analytical method for calculating the properties of the cross section shaped as a solid regular polygon is formulated. For carrying out structural analyses of the cantilever beam, the stress resultants are computed and the differential equation of the elastic curve is derived, in which the effect of shear deformation as well as bending moment is included. The optimal geometries of the strongest beam are determined from the results of the structural analysis. Extreme stress and deflection are chosen as the decision criteria for determining the strongest beam. By using the geometries of the strongest beam obtained in this study, a design example of the minimum weight beam, which can sustain the subjected load with the minimum volume of the beam material, is presented.