Let $(M,\omega)$ carry a Hamiltonian $S^1$-action with moment map $H:M\to\mathbb R$, and let $a\in\mathbb R$ be a regular value of $H$ such that the $S^1$-action on $H^{-1}(a)$ is free. The symplectic cut at level $a$ produces a symplectic manifold \begin{align*} M_{\le a}=\{H<a\}\cup H^{-1}(a)/S^1 \end{align*} whose symplectic form agrees with $\omega$ on the open part $\{H<a\}$ and restricts on the reduced hypersurface to the reduced symplectic form.