Observations are beginning to be sensitive enough to see the outskirts of galaxy clusters, where theory predicts interesting features in the dark matter and gas profiles: the so-called splashback and the accretion shock. I will present our theoretical work in estimating their radial locations using the self-similar spherical collapse model. Our results provide physical understanding to the intriguing behaviours seen in numerical simulations: that the two radii seem to track each other well despite the very different dynamical behaviour of dark matter and gas; and that the properties at cluster outskirts seem to have a different scaling with redshift from that of the inner regions. These findings will guide future observation efforts.