  A graph $G$ is called $H$-saturated if it does not contain any copy of $H$,
but for any edge $e$ in the complement of $G$ the graph $G+e$ contains some
$H$. The minimum size of an $n$-vertex $H$-saturated graph is denoted by
$\sat(n,H)$. We prove $$\sat(n,C_k) = n + n/k + O((n/k^2) + k^2)$$ holds for
all $n\geq k\geq 3$, where $C_k$ is a cycle with length $k$. We have a similar
result for semi-saturated graphs $$\ssat(n,C_k) = n + n/(2k) + O((n/k^2) +
k).$$ We conjecture that our three constructions are optimal.
