- Methods which relate the elastic constants of thermodynamic systems to the fluctuations of the pressure or the volume are called ``fluctuation'' methods. In this paper we derive expressions, within the fluctuation method, for determining elastic constants in systems composed of hard spheres tethered by inextensible bonds. Such systems are frequently used as athermal models of real physical systems whose thermodynamic properties are primarily determined by entropy rather than energy. Our expressions relate the elastic constants to the probability densities of contacts between the spheres and the probability densities of having stretched bonds. We use our formalism to compute, from Monte Carlo simulations, the elastic constants of hard-sphere systems. Our results agree well with analytical predictions and improve the existing numerical data.