We compute the depth and Stanley depth for the quotient ringof the path ideal of length $3$ associated to a $n$-cyclic graph, given some precise formulas for the depth when $n\not\equiv 1\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1\,(\mbox{mod}\ 4)$,and for Stanley depth when $n\equiv 0,3\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1,2\,(\mbox{mod}\ 4)$. We also give some formulas forthe depth and Stanley depth of a quotient of the path ideals of length $n-1$ and $n$.