In the two previous papers of this series we defined a new combinatorial approach to quantum gravity, algebraic quantum gravity (AQG). We showed that AQG reproduces the correct infinitesimal dynamics in the semiclassical limit, provided one incorrectly substitutes the non-Abelian group SU(2) by the Abelian group U(1)3 in the calculations. The mere reason why that substitution was performed at all is that in the non-Abelian case the volume operator, pivotal for the definition of the dynamics, is not diagonizable by analytical methods. This, in contrast to the Abelian case, so far prohibited semiclassical computations. In this paper, we show why this unjustified substitution nevertheless reproduces the correct physical result. Namely, we introduce for the first time semiclassical perturbation theory within AQG (and LQG) which allows us to compute expectation values of interesting operators such as the master constraint as a power series in planck with error control. That is, in particular, matrix elements of fractional powers of the volume operator can be computed with extremely high precision for sufficiently large power of planck in the planck expansion. With this new tool, the non-Abelian calculation, although technically more involved, is then exactly analogous to the Abelian calculation, thus justifying the Abelian analysis in retrospect. The results of this paper turn AQG into a calculational discipline.