This study presents a high-order discontinuous Galerkin (DG) scheme for capturing the Richtmyer–Meshkov Instability (RMI) at polygonal gas interfaces. The two-dimensional, two-component compressible Euler equations are solved using an in-house explicit modal DG solver with third-order accuracy, employing scaled Legendre polynomials and a strong stability-preserving Runge–Kutta time integration method. The accuracy of solver is validated against experimental data for a shocked square light gas interface, demonstrating excellent agreement in interface evolution and characteristic scales. The study focuses on the interaction of shock waves with trapezoidal helium ( \(\text {He}\) ) and sulfur hexafluoride ( \(\text {SF}_{6}\) ) gas interfaces surrounded by nitrogen. Results reveal that density gradients significantly influence the flow evolution, vorticity generation, and instability growth. For the He interface, weaker baroclinic torque leads to gradual vorticity development, moderate vortex roll-up, and a laminar-like evolution with coherent structures. In contrast, the \(\text {SF}_{6}\) interface, with its higher density contrast, exhibits intense baroclinic vorticity generation, amplified Kelvin–Helmholtz instabilities, and strong vortex interactions, resulting in chaotic mixing and turbulent flow behavior. Spatially integrated fields of average vorticity, baroclinic vorticity, and enstrophy provide quantitative insights into the dynamics, showing a sharp increase in instability growth for \(\text {SF}_{6}\) compared to He. The findings highlight the robustness of the high-order DG scheme in accurately capturing complex shock-interface interactions, instabilities, and flow structures.