A plasmonic structure of metal-insulator-metal (MIM) waveguide consisting of a single baffle waveguide and an r-shaped resonator is designed to produce Fano resonance. The finite element method uses the finite element method to analyze the transmission characteristics and magnetic field distributions of the plasmonic waveguide distributions. The simulation results exhibit two Fano resonances that can be achieved by the interference between a continuum state in the baffle waveguide and a discrete state in the r-shaped resonator. The Fano resonances can be simply tuned by changing geometrical parameters of the plasmonic structure. The value variations of geometrical parameters have different effects on sensitivity. Thus, the sensitivity of the plasmonic structure can achieve 1333 nm/RIU, with a figure of merit of 5876. The results of the designed plasmonic structure offer high sensitivity and nano-scale integration, which are beneficial to refractive index sensors, photonic devices at the chip nano-sensors, and biosensors applications.