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The back‐projection algorithm (BPA) is a useful technique for synthetic aperture radar (SAR) imaging. The fast factorised BPA (FFBPA) recursively partitions the back‐projection integral, thus significantly reducing the overall computation complexity corresponding to the improvement obtained by the FFT algorithm compared with the direct implementation of the discrete Fourier transform. In this study, the authors propose a new fast method, termed as the factorised polar‐format BPA (FPFBPA), which combines the polar format algorithm and the factorised back‐projection concept. It is demonstrated that, when the first‐stage subaperture in the FPFBPA contains more than 3 pulses, the proposed method further reduces the computation complexity comparing with FFBPA, provided that the same interpolation method is used. The proposed algorithm is also capable of processing curved orbit and multi‐mode SAR data …