Six edge lengths of a tetrahedron $ (a,b,c,p,q,r) $ are given.
- What relation should be there in order that a tetrahedron can be enclosed among the edge side lengths?
- What is the volume $V$ of the tetrahedron? Is it like $V= \frac{\sqrt 2}{4}\sqrt{abcpqr}?$ The constant is obtained from a regular tetrahedronal special case.
- What is the volume of the circumscribing sphere? Is it like$$ \frac{\pi ~ 64 \sqrt 2}{243 \sqrt 3} \cdot \sqrt{abcpqr}~?$$ The constant is obtained similarly.
Logic is that if any edge length vanishes, then the volumes should also vanish.
Thanking you in advance.