A class of similarity solutions is obtained for radial motions of spherical and cylindrical bodies made of a certain type of compressible hyperelastic materials. The equations satisfied by the infinitesimal generators of the symmetry group of the unified governing first order field equations for spheres and cylinders are found. It is shown that these equations admit a special class of solutions which generate a five-parameter group of transformations. The form of the strain energy function SIGMA corresponding to the resulting symmetry group is evaluated. The similarity variable is determined and ordinary differential equations satisfied by similarity solutions are obtained. Numerical solutions are given for a Ko material which falls into the class of admissible materials.