Let $A$ be a non-trivial abelian group and $A^{*}=Asetminus {0}$. A graph $G$ is said to be $A$-magic graph if there exists a labeling $l:E(G)rightarrow A^{*}$ such that the induced vertex labeling $l^{+}:V(G)rightarrow A$, define by $$l^+(v)=sum_{uvin E(G)} l(uv)$$ is a constant map. The set of all constant integers such that $sum_{uin N(v)} l(uv)=c$, for each $vin N(v)$, where $N(v)$ denotes the set of adjacent vertices to vertex $v$ in $G$, is called the index set of $G$ and denoted by ${rm In}_{A}(G).$ In this paper we determine the index set of certain planar graphs for $mathbb{Z}_{h}$, where $hin mathbb{N}$, such as wheels and fans.