We report a systematic study of cuprous oxide (Cu₂O) films prepared on glass substrates via radio frequency reactive magnetron sputtering at room temperature. We consider the effects of the target power and film thickness on film structures, morphologies, and optical properties. X-ray diffraction indicates significant changes from an amorphous film to a paramelaconite film and then to a cuprite film as the sputtering power increases. Increasing the sputtering power also contributes to variation in the surface roughness of the film. The optical transmittances and bandgaps of the films change with the crystalline structure. The film with a cuprite structure obtained using 60 W of sputtering power exhibits the preferred cuprite orientation (111) with an average roughness of 2.6 nm. This corresponds to an average transmittance of 73% in the 500-1000 nm range and a bandgap of 2.48 eV. The optimized Cu₂O thin film exhibits excellent UV- and blue-light shielding capabilities and high optical transparency in the visible-light region. This suggests possible applications in radiation-proof windows and glasses.