The paper presents the results of study on the accuracy of the geometric shape of shanks (mandrels) of the auxiliary tool in multi-purpose machines using the manufacturing technology with single-axis movement of the cutting tool. Based on the data of mandrel measurements, we constructed the implementations of deviations from the correct geometric shape of the profile curve with an equiaxial contour and found the deviations from the correct geometric shape of the landing surface of the tool mandrels under the action of random and systematic factors within the framework of theory of random stationary processes. It is established that the most rational method of manufacturing mandrels with a profile cross-section curve with an equiaxial contour with the required accuracy of the geometric shape is processing of grinding with a straight flat circle in comparison with the Cup circle. Based on the obtained results, we developed a mathematical model for the geometric shape of the profile connection parts with an equiaxial contour.