We present a simple, analytically solvable magnetohydrodynamics model of current sheet formation through X-point collapse under optically thin radiative cooling. Our results show that cooling accelerates the collapse of the X-point along the inflows, but strong cooling can arrest or even reverse the current sheet elongation in the outflow direction. Hence, we detail a modification to the radiatively cooled Sweet–Parker model developed by Uzdensky & McKinney (Phys. Plasmas, 1962, vol. 18, issue 4, p. 042105) to allow for varying current sheet length. The steady-state solution shows that, when radiative cooling dominates compressional heating, the current sheet length is shorter than the system size, with an increased reconnection rate compared with the classical Sweet–Parker rate. The model and subsequent results lay out the groundwork for a more complete theoretical understanding of magnetic reconnection in regimes dominated by optically thin radiative cooling.