In this study, the stress state of a polygonal plate having a central circular hole with two linear cracks was considered. External forces were applied to the contours of polygonal plate and internal forces were applied to the contours of the circular hole. Conformal mapping function, initially developed by Kuliyev, was used to the make stress analysis of the polygonal plate in physical non-linear statement. As a result, stress concentrations were determined by using conformal mapping function at the end points of cracks. Critical loads where fracture began were defined for different loads. Analytical results of solutions were compared and found to be in agreement with the numerical results in the literature.