A \(\lambda\)-set in a symmetric 2-(v,k,\(\lambda)\) design \(\Pi\) is a subset S meeting every block in 0, 1 or \(\lambda\) points such that for any point of S there is a unique block meeting S at that point only. In this paper, families of symmetric 2-designs containing \(\lambda\)-sets are investigated. First it is shown that if \(\Pi\) has a \(\lambda\)-set, then it is a geometroid with \(v=\lambda u^ 2+u+1\) and \(k=\lambda u+1\), where \(u\geq \lambda -1\). The particular attention is then payed to the existence of three special families of geometroids with \(\lambda\)-sets, namely the geometroids with \(u=\lambda -1\), \(\lambda\), and \(\lambda +1\). Some open problems are discussed, as well.