Random frequency diverse array (RFDA) based directional modulation (DM) was proposed as a promising technology in secure communications to achieve a precise transmission of confidential messages, and artificial noise (AN) was considered as an important helper in RFDA-DM. Compared with previous works that only focus on the spot of the desired receiver, in this work, we investigate a secrecy region around the desired receiver, that is, a specific range and angle resolution around the desired receiver. Firstly, the minimum number of antennas and the bandwidth needed to achieve a secrecy region are derived. Moreover, based on the lower bound of the secrecy capacity in RFDA-DM-AN scheme, we investigate the performance impact of AN on the secrecy capacity. From this work, we conclude that: 1) AN is not always beneficial to the secure transmission. Specifically, when the number of antennas is sufficiently large and the transmit power is smaller than a specified value, AN will reduce secrecy capacity due to the consumption of limited transmit power. 2) Increasing bandwidth will enlarge the set for randomly allocating frequencies and thus lead to a higher secrecy capacity. 3) The minimum number of antennas increases as the predefined secrecy transmission rate increases.