The analysis of thin-walled structures with high slenderness ratios is essential in, e.g., aerospace, energy, and manufacturing applications, where components such as wind turbine blades, pressure vessels, and sheet-metal parts exhibit complex mechanical responses. Accurate numerical modeling is therefore crucial for predicting their behavior under demanding loading and geometric conditions. In this work a fully nonlinear shell formulation based on Reissner-Mindlin kinematics is proposed, employing a six-node displacement-based triangular shell element. Triangular elements offer notable advantages in handling irregular or nonconforming meshes, making them particularly suitable for complex geometries and topologies. The proposed formulation incorporates finite strains, large displacements, and large rotations, ensuring robust performance in highly nonlinear regimes. A key feature of the approach is the re-parameterization of the rotational field using Rodrigues rotation vectors. This representation enables rotations through the superposition of incremental updates, improving numerical stability and accuracy compared to classical Euler-based formulations. The study includes a comparative assessment of the triangular shell element performance across representative nonlinear benchmark problems. A series of numerical analyses demonstrate the reliability of the proposed element in capturing the nonlinear behavior of shell structures under large deformations. Therefore, the resulting nonconforming triangular shell element formulation highlight its suitability for simulations of thin-walled structures involving geometrically intricate structures and complex deformation patterns.