We describe a beam profile monitor design based on Cherenkov light emitted from a charged particle beam in an air gap. The main components of the profile monitor are silicon wafers used to reflect Cherenkov light onto a camera lens system. The design allows for measuring large beam sizes, with large photon yield per beam charge and excellent signal linearity with beam charge. The profile monitor signal is independent of the particle energy for ultrarelativistic particles. Different design and parameter considerations are discussed. A Cherenkov light-based profile monitor has been installed at the FACET User Facility at SLAC. We report on the measured performance of this profile monitor.