The Drucker-Prager（D-P） criterion

which is used to approximate the Mohr-Coulomb（M-C） hexagon with a circle on the π plane and adopted by most general numerical analysis programs

has been researched by using the least square approximate theory.A close-form formulation about the square error has been gained.At the same time

an optimal D-P criterion

which corresponds to the minimum square error

has been proposed.The analysis of square error indicated that the D-P criterion matched with the compressive meridian of the M-C generates more errors than others

and the square error of the equivalent area criterion is the closest to that of the optimal criterion

as well as the rest are nearly same.Furthermore

according to the curves of reduction factor（RF） vs.iterative number gained by strength reduction analysis

it can be found that the curves of the equivalent area criterion and the optimal criterion are closer to that of the M-C criterion when the system is in stable condition.However

the curve of the M-C criterion will approaches the inscribed circle when the system is in unstable condition.