 The possible usage of different carbon nanostructures, including C24 nanocage, carbon nanotube (CNT), and hexa-peri-hexabenzocoronene nanographene (HBC) is studied in the anode of K-ion batteries (KIBs) by DFT calculations. Based on the calculations, both of K and K+ species preferentially adsorb on the center of a hexagon of the studied nanostructures. The results show that the underlying mechanism of the K+ adsorption is the cation-π interaction, and by increasing the curvature of the adsorbents the interaction becomes weaker. When the curvature is reduced, the adsorption energy of the K atom becomes more positive and the adsorption mechanism changes from ionic to nonionic. This study suggests that the mechanism of the strength of K interaction and the structure curvature are the main factors in determining the cell voltage (Ucell) of the KIBs. The C24 cage generates a negative Ucell which is physically meaningless, and it cannot be used as an anode material. The CNT produces a very small Ucell of 0.09 V, while the HBC is suggested to be a proper nanostructure for use in the KIBs because of its large Ucell about 1.45 V.