Summary: The volume of the hive polytope (or polytope of honeycombs) associated with a Littlewood-Richardson coefficient of \(\mathrm{SU}(n)\), or with a given admissible triple of highest weights, is expressed, in the generic case, in terms of the Fourier transform of a convolution product of orbital measures. Several properties of this function – a function of three non-necessarily integral weights or of three multiplets of real eigenvalues for the associated Horn problem – are already known. In the integral case it can be thought of as a semi-classical approximation of Littlewood-Richardson coefficients. We prove that it may be expressed as a local average of a finite number of such coefficients. We also relate this function to the Littlewood-Richardson polynomials (stretching polynomials) i.e. to the Ehrhart polynomials of the relevant hive polytopes. Several \(\mathrm{SU}(n)\) examples, for \(n = 2, 3,\ldots, 6\), are explicitly worked out.