IceCube-Gen2, the next generation extension of the IceCube Neutrino Observatory at the South Pole, offers a unique scientific potential for cosmic-ray physics at PeV to EeV energies complementing the main science case of neutrino astronomy. The cosmic-ray science case will be enabled by a surface array on top of an extended optical array deep in the polar ice. The optical array measures TeV muons of air showers, and the surface array primarily measures the electromagnetic shower component and low-energy muons. The design of the surface array foresees scintillation panels providing a full-efficiency threshold for near-vertical proton showers of 0.5 PeV and radio antennas increasing the measurement accuracy for the electromagnetic shower component in the energy range of the Galactic-to-extragalactic transition. Compared to IceCube, the aperture for air showers measured in coincidence with the surface and optical arrays will increase by a factor of 30, due to the larger area and angular acceptance in zenith angle. The science potential includes both, the particle physics of air showers, such as prompt muons, and the astrophysics of the highest energy Galactic cosmic-rays, enabled by the higher sensitivity for the mass composition and anisotropy of cosmic rays, and by the search for PeV photons. This proceeding summarizes the science case and design of the surface array as presented in the recently released IceCube-Gen2 Technical Design Report: https://icecube-gen2.wisc.edu/science/publications/tdr/