The only remaining nontrivial case is the pentagonal base, and it's clear that all the needed heights are no more involved than stuff containing sqrt(5).  For the hexagonal base, the pyramid is flat, zero volume, and for higher order polygons, the lateral edges cannot reach the vertex while remaining of unit length.   --   Gene From: Don Reble <djr@nk.ca> To: math-fun@mailman.xmission.com Sent: Friday, September 23, 2016 1:52 PM Subject: Re: [math-fun] Bar bet

"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges." True or false?  --rwg

    Trick puzzle: prove that the volume of _every_ such pyramid     (unit edges, regular polygon base) is a quadratic surd. -- Don Reble  djr@nk.ca