Aiming at simplifying the hardware structure and reducing the energy consumption in wireless communication via reconfigurable intelligent surfaces (RIS), this paper introduces a novel RIS design founded on the correlation between the phase shift values of the surface elements. First, a correlation analysis is conducted, considering the azimuth angle of a target device within a coverage region spanning from $-80^{\circ}$ to $80^{\circ}$. The correlation is demonstrated for different deployment cases, creating the basis for the new RIS structure, termed Connected-RIS, where correlated elements are designed to share the same control signal. The fundamental performance of the proposed design is then analyzed in terms of control signals, power consumption, and communication system performance, comparing it to two RIS structures with full control: one with the same size as the proposed design, and the other employing the minimum number of elements necessary to satisfy the fair coverage criterion. The correlation-based RIS design enables three-dimensional passive beamforming and significantly reduces the number of required load impedances and control signals, thereby lowering the hardware cost and simplifying the control circuitry. It also achieves substantial power savings as compared to the baseline schemes, while maintaining sufficient gain for a fair radio coverage. For instance, numerical simulations demonstrate that the proposed design reduces the power consumption by almost 86-92\% and the control signals by 83-98\% compared to operation with fully controlled RIS.