It is shown that the optimum wall thickness of a cylindrical round tube column is a function of load only and is independent of diameter. The optimum wall thickness of a tapered round thin-walled column is found to be constant along its length. The optimum shape of a tapered round thin-walled column is derived, being that column whose bending stress in the buckled state is constant along its length. The weight ratio of the optimum tapered column to an equal-strength optimum cylindrical column is found to be 0.8924. It is shown that a double truncated cone whose diameter ratio is in the range 0.35 ⩽ D1/D2 ⩽ 0.50 closely approaches the optimum column. If it is specified that no portion of the double truncated cone shall yield, then the weight advantage of the cone over the cylindrical column is rapidly lost as the stress in the cylindrical column approaches the yield stress. In the inelastic range the weight advantage of the tapered column will be less than in the elastic range.