Enumerationofperfectmatchingsongraphshasalongstandinginterestincombinatorialmathematics.Inthispaper,weobtainsomeexplicitexpressionsofthenumberofperfectmatchingsforatypeofArchimedeanlatticeswithtoroidalboundarybyapplyingTesler'scrossingorientationstoobtainsomePfaffianorientationsandenumeratingtheirPfaffians.