This paper analyzes the Ergodic Capacity (EC) as a key performance metric for reconfigurable intelligent surface (RIS)-aided communication channels. We classify these channels into Centralized RIS (C-RIS) and Distributed RIS (D-RIS) configurations. A primary challenge in this area is deriving tractable closed-form EC expressions, especially for D-RIS, as existing methods often yield complex results that are impractical for system design. To address this gap, we propose three novel approximations. First, we derive a precise closed-form EC expression for D-RIS systems using Meijer-G functions, which also serves as a general framework that includes C-RIS as a special case. Second, we introduce two efficient numerical approximations based on the Gauss-Laguerre method, providing an excellent balance between high accuracy and low computational complexity. All proposed methods are rigorously validated against Monte Carlo simulations. Furthermore, using these approximations, we provide new insights into how system parameters, such as the number of RIS elements and transmit power, directly impact the achievable EC.