Does there exists 'Superhero' cuboids?

Question

There exists heronian figures,i.e., the figures with integral area and perimeter. This also works for heronian 3D shapes, which have integral volume and integral surface area. Some figures, which are actually special have the same area and perimeter. They are informally referred to as superhero figures. But, my question is simply, does there exist any superhero cuboids, where the volume and the surface area are the same(in any units)? If so, how can we prove it? Link to superhero triangles: Superhero triangles

Answer 1

If $a$, $b$, $c$ are the (integer) edges of the cuboid, surface area and volume are equal if $2(ab+bc+ca)=abc$, that is if: $$ {1\over a}+{1\over b}+{1\over c}={1\over 2}. $$ From there, it is not difficult to find all possible cases:

$$ \begin{array}{|c|c|c|} \hline a & b & c \\ \hline 3 & 7 & 42\\ \hline 3 & 8 & 24\\ \hline 3 & 9 & 18\\ \hline 3 & 10 & 15\\ \hline 3 & 12 & 12\\ \hline 4 & 5 & 20\\ \hline 4 & 6 & 12\\ \hline 4 & 8 & 8\\ \hline 5 & 5 & 10\\ \hline 6 & 6 & 6\\ \hline \end{array} $$