  In the search for axionic Dark Matter, the high frequency part of the 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 angle polygonal prisms enclosed within two flat endcaps. For the
axion-sensitive (pseudo-)TM010 mode, finite element simulations yield form
factor of the order of 0.8 and Q factor of the order of 60000 for a copper
cavity at 4$\,$K. High tunability of up to $\sim 5\%$ is achieved by reciprocal
rotation of the two prisms, without significant changes in haloscope
sensitivity. A prototype aluminium hexagonal cavity was built and tested,
confirming the main characteristics of the design.
