Summary: Surface artefacts are features in a surface which cannot be avoided by movement of control points. They are present in B-splines, box splines and subdivision surfaces. We showed how the subdivision process can be used as a tool to analyse artefacts in surfaces defined by quadrilateral polyhedra [the authors, Lect. Notes Comput. Sci. 3604, 350‒363 (2005; Zbl 1141.68639); Comput. Aided Geom. Des. 28, No.~3, 177‒197 (2011; Zbl 05909583)]. In this paper, we are utilising the subdivision process to develop a generic expression which can be employed to determine the magnitude of artefacts in surfaces defined by any regular triangular polyhedra. We demonstrate the method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes: loop subdivision, butterfly subdivision and a novel interpolating scheme with two smoothing stages. We compare our results for surfaces defined by triangular polyhedra to those for surfaces defined by quadrilateral polyhedra.