Formulas for the backscattering and total scattering cross sections for dielectric coating spheres of arbitrary size were developed following the methods of Hansen and of Aden and Kerker. Calculations based on the resultant formulas were made in the region of the first resonance (outside dimension of the concentric spheres is approximately 0.4 wavelength). It was found that certain choices of coating thickness and dielectric constant can lead to increases in the scattering cross sections of more than 100 percent relative to a solid metal sphere of the same outside diameter. A number of calculated and measured curves are presented, and they show the variation of the backscattering and total scattering cross sections as functions of coating thickness, coating permittivity, and outside diameter of the coating. Where both calculations and measurements were made for the same conditions, the values agree to ±2 percent.