Which point sets admit a k-angulation?

For k ≥ 3, a k-angulation is a 2-connected plane graph in which every internal face is a k-gon. We say that a point set P admits a plane graph G if there is a straight-line drawing of G that maps V (G) onto P and has the same facial cycles and outer face as G. We investigate the conditions under which a point set P admits a k-angulation and find that, for sets containing at least 2k2 points, the only obstructions are those that follow from Euler’s formula.