Electromagnetic backscatter characteristics of conducting polyhedral spheres, constructed of 12 pentagons and a large number of hexagons, were investigated. The polyhedral structures were classified by their number of vertices and by the width of the conducting edges in their frames. Polyhedral spheres with 60, 80, 180, 240, 320, 500, 540, 960, and 1500 vertices were evaluated for their ability to scatter electromagnetic waves back to a source. The edge widths were varied between the limits of (1) nearly zero, to produce a wire-frame sphere; and (2) full surface closure of the polygon holes in the sphere, to form solid conducting structures. Using Method-of-Moments simulations with the WIPL-D code, the monostatic scattering properties of these spheres were studied over a wide range of structural and frequency parameters. The spherical wire frames had up to a 10 dB larger radar cross section than the completely solid spheres. This result provides models for new light-weight radar cross section targets used as satellite-calibration spheres with enhanced radar cross section for easier detection, low mass for easier launch, and low drag cross section for long orbit life. With selected edge widths, the monostatic radar cross section becomes vanishingly small at specific anti-backscatter frequencies, for conducting spherical shells with a regular pattern of polygon holes. These shells can be placed around solid spheres to nearly eliminate radar reflection from the composite object at specific frequencies. The polyhedral conducting-mesh (PCM) spheres with vanishing radar cross section at a desired frequency have applications for isotropic scattering sources that reflect no energy back to the electromagnetic source. the anti-backscatter polyhedral conducting-mesh spheres may provide insight for future stealth applications with spherical targets.