Let G = (V,E) be a graph of order p and size q. It is known that if G is super edge-magic graph then q 2p−3. Furthermore, if G is super edge-magic and q = 2p−3, then the girth of G is 3. It is also known that if the girth of G is at least 4 and G is super edge-magic then q 2p − 5. In this paper we show that there are infinitely many graphs which are super edge-magic, have girth 5, and q = 2p−5. Therefore the maximum size for super edge-magic graphs of girth 5 cannot be reduced with respect to the maximum size of super edge-magic graphs of girth 4.