Abstract A fullerene graph G is a plane cubic graph such that every face is bounded by either a hexagon or a pentagon. A set H of disjoint hexagons of G is a resonant set (or sextet pattern) if G−V(H) has a perfect matching. A resonant set is a forcing set if G−V(H) has a unique perfect matching. The size of a maximum resonant set is called the Clar number of G. In this paper, we show the Clar number of fullerene graphs with a non-trivial cyclic 5-edge-cut is (n−20)/10. Combining a previous result obtained in Kardoš et al. (2009), it is proved in this paper that a fullerene has the Clar number at least (n−380)/61. For leapfrog fullerenes, we show that the Clar number is at least n/6−n/5. Further, it is shown that the minimum forcing resonant set has at least two hexagons and the bound is tight.