In this paper, the optimized design of a rectangular O-ring groove has been analyzed for a maximum Cauchy stress and maximum strain using the Taguchi method. This method may efficiently optimize the design parameters for an O-ring groove of a LPG filling unit. The computed FEM results indicate that the optimized design parameters can only be drawn by nine experimental numbers of iterations when the Taguchi design technique has been employed with a finite element method. This means that the Taguchi design method is very useful for the optimization design of O-ring rectangular groove geometry. Based on the computed FEM results by the Taguchi design technique, the dimensions of a groove geometry are given as h=2.5 mm, d=2.74 mm, c=0.15 mm, and w=3.0 mm. In this study, the initial compression ratio of O-rings is recommended as 8.7% for a gas supply pressure of 18 $kg/cm^2$.