This paper analyzes the information theoretic secrecy performance in finite-area wireless networks based on a stochastic geometry framework. Unlike most prior works, which explored the physical layer security with a large number of transmitters, legitimate receivers and eavesdroppers in infinite regions, we consider a finite downlink wireless network composing of a transmitter, a legitimate receiver and several eavesdroppers. The legitimate receiver attempts to receive confidential data from the transmitter in the presence of the eavesdroppers. We present the probabilistic characteristics of the achievable secrecy rates and average secrecy rates in both disk regions and regular -sided convex polygon regions. As shown by extensive numerical results, the proposed framework could be leveraged to efficiently analyze the secrecy performance of finite-area networks, and give insights for network designers on how to achieve good secrecy performance in finite-area networks.