Annular acoustic black holes (ABHs) have been recently proposed as a potential means for reducing vibrations of cylindrical shells. The latter are very common structures in the naval, aeronautic and industrial sectors so widening ABH applications from flat plates to curved structures seems worth exploring. This work focuses on the benefits of embedding annular ABH indentations on cylindrical shells to reduce outward sound radiation. The goal of the paper is to propose a semi-analytical method to determine the acoustic power, radiation efficiency, source location and far-field acoustic pressure of ABH shells and compare them with those of uniform thickness shells. The vibration field of the ABH and uniform cylindrical shells is computed by means of the Gaussian expansion method (GEM) within the Rayleigh–Ritz approach. Then, the radiated pressure is obtained by solving the Helmholtz equation in cylindrical coordinates using the Green's function method. The surface of the cylinder is discretized into small finite size radiators and an impedance matrix is used to obtain the acoustic surface pressure in terms of the shell radial velocity. To determine those regions of the cylinder responsible for the far field radiated sound, use is made of supersonic sound intensity (SSI). A method is proposed to calculate the SSI in the spatial domain for cylindrical shell structures which allows one to make direct use of the previously computed surface pressure and velocity distributions. The whole methodology is validated against finite element method (FEM) simulations and after that, results are presented for an acoustically thick shell. The roles played by the critical and ring frequencies are reported and the spectra of the acoustic power, radiation efficiency and far field acoustic pressure get analysed. It is shown that the annular ABH can become very effective when the cylinder flexural motion dominates over the circumferential one. The slow down of bending waves inside the ABH makes structural supersonic waves (in relation to sound speed) become subsonic at some point, which clearly diminishes the shell radiation efficiency. Overall, it is described why embedding an annular ABH on a cylindrical shell can strongly help reducing the radiated sound to the far-field.