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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.

Air Mike
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Narasimham
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    I will assume $p,q,r$ corresponds to edges opposite to $a,b,c$. They are realizable as edge lengths of tetrahedron iff 1) the faces satisfy triangle inequalities AND 2) the corr Cayler Menger determinant is positive. see refs in this answer. The volume can be computed using the CM determinant. The formula for circumradius can be found in this question – achille hui Feb 17 '24 at 12:06
  • Thanks@ achille hui for so many links/references. Actually verified given probable guess cases to be incorrect before posting itself. Often wonder about guessing... how generalizations are done from particular cases if not using prior knowledge ! :) – Narasimham Feb 18 '24 at 01:49

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