The elastic moduli of composite materials consisting of an isotropic, elastic matrix with perfectly rigid inclusions have been studied near random close packing. In two dimensional systems of uniform sized disks, the moduli have been determined by computer simulations. A theory of the bulk modulus k is developed by assuming that the elastic energy of the neck regions is minimized subject to the constraint that average local strain, <εi >, equals the macroscopic strain. Here the change in center-to-center distance to the ith nearest neighbor disk is δri =εi ri . We show that εi ∝ √ wi , where wi is the gap (ri −2R, R=disk radius). This prediction has been verified by the simulations, which also confirm our theory. Our findings appear to be the first example of local geometry dominating the strain near close packing. Predictions for the bulk modulus K in three dimensional systems of rigid spheres (radius = R) are also made. In this case, εi ∝ 1/log(R/wi ).