We designed a high sensitivity plasmonic sensor based on a metal–insulator–metal bus waveguide coupled with a triangular ring-shaped resonator (TRSR) containing silver nanorods working in the mid-infrared region having multiple modes. We applied the two-dimensional finite element method (FEM) for transmittance, electric and magnetic fields, plasmon modes, and sensing performance investigation. The highest sensitivity for aqueous solutions (1.31≤n≤1.41) is relevant for extended applications to biosensing, while geometrical optimization is still possible for higher sensitivity. The developed structure receives the high sensitivity and figure of merit ranging in 1000∼6000 nm/RIU (refractive index unit) and 11.67∼66.67 1/RIU for modes 1–7, respectively. Moreover, the designed structure can serve as the refractive index biosensors, significantly enhancing the sensitivity by 300% compared to its common one.