Abstract A method is presented to solve the characteristic impedance of transmission lines of specific cross-section. In this method, we use the circle or ellipse to approximate polygonal boundaries, taking account of ′angular effect′ of the electric charge of polygonal outer conductor, and take geometric mean of the upper and lower bounds to the size of the cross-section. The formulae for calculating the characteristic impedance of five new transmission lines (i.e. with rectangular outer and elliptic inner, circular outer and equilateral triangular inner, circular outer and regular pentagonal inner, circular outer and regular hexagonal inner, elliptic outer and rectangular inner cross sections respectively and nine existing transmission lines are obtained, all expressed in terms of elementary functions. The accuracy of the expressions is confirmed by comparing the numerical values obtained with exact values of same relative problems in the literatures.