Let P n be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that |P n |=<n!(75.8) n + o ( n ) and Bender, Gao and Wormald proved that |P n |>=n!(26.1) n + o ( n ) . Gerke and McDiarmid proved that almost all graphs in P n have at least 13/7n edges. In this paper, we show that |P n |=<n!(37.3) n + o ( n ) and that almost all graphs in P n have at most 2.56n edges. The proof relies on a result of Tutte on the number of plane triangulations, the above result of Bender, Gao and Wormald and the following result, which we also prove in this paper: every labelled planar graph G with n vertices and m edges is contained in at least 3 ( 3 n - m ) / 2 labelled triangulations on n vertices, where is an absolute constant. In other words, the number of triangulations of a planar graph is exponential in the number of edges which are needed to triangulate it. We also show that this bound on the number of triangulations is essentially best possible.