The effect of oblique angle on the localized surface plasmon resonance of cone-shaped nanoparticles is explored. The finite difference time domain method is applied to analyze the extinction spectra and the electric field intensity. The simulation results show that cone nanoparticles with a small oblique angle exhibit a larger resonant wavelength than those with a large oblique angle. The full width at half maximum initially decreases and then increases as the oblique angle increases, with a turning point of 17.5 degrees. The refractive index sensitivity and the sensing sensitivity both decrease when the oblique angle of the cone nanoparticles increases, achieving a maximum figure of merit (FOM) of 14.6 RIU(-1). For cone-shaped nanoparticles, the shape can be described by the power exponent m of the curve in the cross-section. The larger the power exponent m, the larger the oblique angle. Cone-shaped nanoparticles with power exponent m < 1 (concave conical surface) exhibit larger FOM than those with m > 1 (convex conical surface). The largest FOM is achieved when m = 1/3.