A new upper-bound solution is developed for the extrusion of polygonal sections through shaped dies. The surface of the die is generated by an envelope of curved lines drawn from points on the perimeter at the entry section to corresponding points at the exit of the die. The present investigation is concerned with the analysis of streamlined dies (a subset of shaped dies), though the method is applicable to other curved dies such as parabolic, convex or concave shapes. It is concluded that the streamlined die is superior to the straightly converging dies from the aspect of reducing the extrusion load.