We calculate for the first time the equation of state and other thermodynamic properties of the planar square-well dumbbell (SWD) fluid. The SWD molecule consists of two fused hard discs of diameter sigma, separated by a reduced separation L* = L/sigma with at tractive annuli of outer radii, Ksigma around each hard disc. Monte Carlo simulations were carried out over a range of densities and temperatures for two choices of potential width and several molecular elongations. In particular, we concentrated on states with the molecular elongation parameter, L* = 0.3 and the potential-width parameter, K = 1.5. In this article we concentrate on analytic representations of the thermodynamic behavior of the SWD fluid. The second virial coefficients of the SWD system are calculated by a part analytic/part numerical procedure. The free energy expansion coefficients are calculated up to third order. Convergence of the perturbation expansion based on the hard-dumbbell fluid is examined and shown to be very satisfactory. A generalization of the quasichemical (QCA) coordination number model for the SWD fluid is proposed. The equation of state and energy derived from this model, compared with the simulation results, show that the QCA model represents well the simulated properties of the SWD at low density but fails at high density.