In the search for axionic dark matter, the high-frequency part of the quantum chromodynamics (QCD) axion parameter space is favored, as indicated by both cosmological and astrophysical arguments and recent indications from lattice QCD calculations. To extend the probing range of cavity haloscopes, solutions addressing the unfavorable scaling of cavity volume with frequency must be developed. Here, we present a novel type of high-volume thin-shell resonator for high-frequency haloscope dark-matter searches. The cavity is formed by two nested and coaxial right-angled polygonal prisms enclosed within two flat endcaps. For the axion-sensitive (pseudo-)TM010 mode at about 10 GHz, finite-element simulations yield a form factor on the order of 0.8 and a Q factor on the order of 60 000 for a copper cavity at 4 K. Reasonable tunability of up to ∼5% is achieved by reciprocal rotation of the two prisms, without relevant changes in haloscope sensitivity. A prototype aluminum hexagonal cavity was b...