A classical 6j-symbol is a real number which can be associated to a labelling
of the six edges of a tetrahedron by irreducible representations of SU(2). This
abstract association is traditionally used simply to express the symmetry of
the 6j-symbol, which is a purely algebraic object; however, it has a deeper
geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a
striking (but unproved) asymptotic formula relating the value of the 6j-symbol,
when the dimensions of the representations are large, to the volume of an
honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal
of this paper is to prove and explain this formula by using geometric
quantization. A surprising spin-off is that a generic Euclidean tetrahedron
gives rise to a family of twelve scissors-congruent but non-congruent
tetrahedra.