In order to meet the design needs of gear pairs with undercut in external gear pumps,based on the machining principle of gear hobbing, from creating a calculation radius formula of gear undercut circle, a complete set of calculation formula of spur gear contact ratio factor suitable for gear pairs with undercutis derived. Based on this formula, the influence of gear basic parameters to the spur gear contact ratio factor with undercut was analyzed; finally, all theobtainedresults wereverified by threedimension (3D)virtual model of spur gear pairs with undercut. The results show that, because of the gear undercut, the apex of driving gear tooth maybe excluded from being meshed, the common calculation method of spur gear apex contact ratio factor instead of spur gear undercut contact ratio factoris not reliable at present.The spur gear undercut contact ratio factor alwaysis affected bygear basic parametersexcept gear module, among them, the influence of gear modification coefficient is the largest. Spur gear undercut contact ratio factor canbeensuredto equal to spur gear apex contact ratio factor by employing the best rack tool tipradius.When the actual spur gear undercut contact ratio factoris too large,the transmission quality of gear pairs was improved by reducing the rack tool tipradius,and etc. It is concluded that a new method for the other conventional undercutgear pairs designis providedby the complete calculation formula of spur gear undercut contact ratio factor.