Abstract The torsion of bars possessing a regular polygonal cross-section by means of boundary collocation method is considered herein. Seven cases of bars are considered: (1) regular polygonal bars, (2) regular polygonal bars with circular centred holes, (3) cylindrical bears with regular polygonal centred holes, (4) regular polygonal bars with regular polygonal centred holes, (5) regular polygonal bars with circular centred reinforced rod, (6) cylindrical bars with regular polygonal reinforced rod, (7) regular polygonal bars with regular reinforced rod. By means of analytical integration of the stress functions for the seven cases considered, the analytical formulae for the non-dimensional stiffness of the bars are obtained. Numerical results for different cross-sectional shapes are presented to demonstrate the efficiency and accuracy of the method. The results obtained should be useful in the design of torsion members.