Polyhedral silica nanofoam (PNF-SiO2) analogues of dry soap froths with minimal surface area were used as templates for making polyhedral carbon nanofoams (PNF-C). Furfuryl alcohol or triblock copolymers were used as carbon sources. The volume of carbon precursor relative to the internal pore volume of PNF-SiO2's was systematically varied between 50% and 100% in order to investigate the effect of filling fraction on internal structure of the corresponding PNF-C's. Transmission electron microscopy, small-angle X-ray scattering and nitrogen physisorption were used to characterize the samples. To aid the interpretation of the experimental data, a model for X-ray scattering from spherical shells was used to approximate scattering from the polyhedral foam cells. PNF-C's cast from the PNF-SiO2's, displayed the characteristic Plateau borders of minimal surface area foams defining interconnected, slit-like pore systems at all filling fractions. At relatively high filling fractions, inverse foam structures were obtained with the slit-like pores systems interpenetrating aggregated, close-packed, relatively low density polyhedral carbon nanoparticles co-joined by carbon struts. At relatively low filling fractions, polyhedral carbon nanofoams with relatively thin, fused double-wall structures and interconnected polyhedral pore systems were obtained.