Volume of Smallest Rational Tetrahedron

Theorem

The only rational tetrahedron whose edge lengths are less than $157$ has:

edges of length $117$, $80$, $53$, $52$, $51$, $84$

faces of area $1800$, $1890$, $2016$, $1170$

volume of $18 \, 144$.

Proof

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Sources

• 1992: R.H. Buchholz: Perfect Pyramids (Bull. Austral. Math. Soc. Vol. 45: pp. 353 – 368)

• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $18,144$

• Weisstein, Eric W. "Heronian Tetrahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HeronianTetrahedron.html