In this paper, to further improve the coverage and performance of unmanned aerial vehicle (UAV) communication systems, we propose a reconfigurable intelligent surface (RIS)-assisted UAV scheme where an RIS installed on a building is used to reflect the signals transmitted from the ground source to an UAV, and the UAV is deployed as a relay to forward the decoded signals to the destination. To model the statistical distribution of the RIS-assisted ground-to-air (G2A) links, we develop a tight approximation for the probability density function (PDF) of the instantaneous signal-to-noise ratio (SNR). By using the obtained distribution, analytical expressions for the outage probability, average bit error rate (BER), and average capacity are derived. Results show that the use of RISs can effectively improve the coverage and reliability of UAV communication systems.