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Since Compton cameras were introduced in the use of single photon emission computed tomography, various types of conical Radon transforms, which integrate the emission distribution over circular cones, have been studied. Most of previous works did not address the attenuation factor, which may lead to significant degradation of image quality. In this paper, we consider the problem of recovering an unknown function from conical projections affected by a known constant attenuation coefficient called an attenuated conical Radon transform. In the case of a fixed opening angle and vertical central axis, new explicit inversion formula is derived. Two‐dimensional numerical simulations were performed to demonstrate the efficiency of the suggested algorithm. [ABSTRACT FROM AUTHOR]