A new approach for constructing polar-like boundary-conforming coordinates inside a toroid with strongly shaped cross-sections is presented. A coordinate mapping is obtained through a variational approach, which involves identifying extremal points of a proposed action in the mapping space from [0, 2π] 2 × [0, 1] to a toroidal domain in R 3. This approach employs an action built on the squared Jacobian and radial length. Extensive testing is conducted on general toroidal boundaries using a global Fourier-Zernike basis via action minimisation. The results demonstrate successful coordinate construction capable of accurately describing strongly shaped toroidal domains. The coordinate construction is successfully applied to the computation of three-dimensional magnetohydrodynamic equilibria in the GVEC code where the use of traditional coordinate construction by interpolation from the boundary failed.