The synthetic aperture focusing technique (SAFT) has been shown to significantly improve the lateral resolution of monostatic imaging systems. To apply SAFT to interior imaging of cylindrical structures such as pipes or blood vessels, algorithms adapted to cylindrical measurement surfaces are needed. In this paper, we present a new algorithm for SAFT applied to such cylindrical scans. The derivation of the algorithm is based directly on the Fourier domain solutions to the wave equation in cylindrical coordinates, and in this sense, the algorithm is exact. Compared with existing methods, the proposed algorithm yields lower side lobes and better resolution for wide-beam transducers. The attainable angular resolution is shown to depend on the size of the transducer and the scanning radius R. For a flat, circular transducer of diameter D, the angular resolution is approximately D/(2R). It is also shown that using an angular step size of D/(4R) will keep grating lobes at least 40 dB below the main lobe, and we therefore recommend D/(4R) as a suitable step size for practical applications.