We study straight-line drawings of planar graphs with prescribed face areas. A plane graph is 'area-universal' if for every area assignment on the inner faces, there exists a straight-line drawing realizing the prescribed areas. For triangulations with a special vertex order, we present a sufficient criterion for area-universality that only requires the investigation of one area assignment. Moreover, if the sufficient criterion applies to one plane triangulation, then all embeddings of the underlying planar graph are also area-universal. Until now, it is open whether area-universality is a property of a plane or planar graph. We use the developed machinery to present area-universal families of triangulations. Among them we characterize area-universality of accordion graphs showing that area-universal and non-area-universal graphs may be structural very similar.