Estimating the crack growth path in rock masses is one of the important issues in many mining and civil engineering projects such as hydraulic fracturing, mine drilling, tunnel construction, etc. Estimating the crack growth path is generally considered from two perspectives: first, estimating the crack growth path to prevent its progression, and second, controlling the crack growth in the desired direction to facilitate the failure of rock masses. For this purpose, a suitable method should be presented to estimate the crack growth path as accurately as possible. The aim of this paper is to present a simple but accurate method for estimating the crack growth path in cracked rock fragments under compound loading. The proposed method is a step-by-step method in which at each stage the fracture initiation angle is calculated based on the maximum tangential stress criterion, taking into account the effects of higher-order terms. Finally, the proposed method is evaluated by the laboratory results reported in previous papers to verify its accuracy and precision. The results of the studies show that if the three terms of the Williams series expansion are used to calculate the fracture initiation angle at each stage, the proposed method can predict the crack growth path of rock fragments with good accuracy.