is there any way to calculate height of irregular tetrahedron with pythagoras theorem

Question

given an irregular tetrahedron of side length $a$,$b$,$c$,$d$,$e$,$f$, is it possible to find the perpendicular distance from its apex to base with the help of pythagoras theorem

Answer 1

Yes, using two expressions for the volume of the tetrahedron $ABCD$ \begin{align} V&=\tfrac13\,S_{abc}\cdot h ,\\ V&= \frac1{12}\, \left( 4\, u^2\, v^2\, w^2+(u^2+v^2-c^2)\, (v^2+w^2-a^2)\, (u^2+w^2-b^2) \right. \\ &\phantom{=} \left. -u^2\, (v^2+w^2-a^2)^2-v^2\, (u^2+w^2-b^2)^2-w^2\, (u^2+v^2-c^2)^2 \right)^{1/2} , \end{align}

where

\begin{align} a&=|BC| ,\quad b=|AC| ,\quad c=|AB| ,\\ u&=|AD| ,\quad v=|BD| ,\quad w=|CD| . \end{align}

See The volume of the pyramid in terms of its side lengths for the reference.