This paper describes a new method for obtaining optimal die shape which produces minimal stress in the extrusion or the drawing of nonaxisymmetric sections from round bar stock. The surface of the die is generated by an envelope of straight lines drawn from the points on the perimeter at the entry section to the corresponding points at the exit of the die. An upper bound technique based on the kinematically admissible velocity field is used to determine the forming stress for different regular polygonal sections (the number of sides is from three to infinite) for a given reduction of area, material property, and frictional condition. The theoretical results are compared with known experimental and theoretical data and are found to be in good agreement.