  Quantum polyhedra constructed from angular momentum operators are the
building blocks of space in its quantum description as advocated by Loop
Quantum Gravity. Here we extend previous results on the semiclassical
properties of quantum polyhedra. Regarding tetrahedra, we compare the results
from a canonical quantization of the classical system with a recent wave
function based approach to the large-volume sector of the quantum system. Both
methods agree in the leading order of the resulting effective operator (given
by an harmonic oscillator), while minor differences occur in higher
corrections. Perturbative inclusion of such corrections improves the
approximation to the eigenstates. Moreover, the comparison of both methods
leads also to a full wave function description of the eigenstates of the
(square of the) volume operator at negative eigenvalues of large modulus.
  For the case of general quantum polyhedra described by discrete angular
momentum quantum numbers we formulate a set of quantum operators fulfilling in
the semiclassical regime the standard commutation relations between momentum
and position. Differently from previous formulations, the position variable
here is chosen to have dimension of (Planck) length squared which facilitates
the identification of quantum corrections. Finally, we provide expressions for
the pentahedral volume in terms of Kapovich-Millson variables.
