The cactus graph has many practical applications, particularly in radio communication systems. Let $G=(V,E)$ be a finite, undirected, and simple connected graph, then the edge metric dimension of $G$ is the minimum cardinality of the edge metric generator for $G$ (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices ${\mathcal{G}}_e=\{g_1,g_2,...,g_k\}$ of a connected graph $G$, for any edge $e\in E$, we referred to the $k$-vector (ordered $k$-tuple), $r(e|{\mathcal{G}}_e)=(d(e,g_1),d(e,g_2),...,d(e,g_k))$ as the edge metric representation of $e$ with respect to $G_e$. In this regard, ${\mathcal{G}}_e$ is an edge metric generator for $G$ if, and only if, for every pair of distinct edges $e_1,e_2\in E$ implies $r(e_1|{\mathcal{G}}_e)\ne r(e_2|{\mathcal{G}}_e)$. In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: $\mathfrak{C}(n,c,r)$ and $\mathfrak{C}(n,m,c,r)$ in terms of the number of cycles $(c)$ and the number of paths $(r)$.