<p>This study presents a novel approach for analyzing and calculating single-point cutting tool angles using the mathematical framework of geometric algebra. The relationship between the tool angles in the static and working states can be derived by transforming the coordinate systems attached to the tool in Euclidean space. These transformations are represented as rotations using spinors, with rotation angle parameters that encode essential information regarding the tool angles. Moreover, defining tool surfaces as bivectors further enhances modeling capabilities. As fundamental concepts in geometric algebra, spinors and bivectors encapsulate profound geometric interpretations within a concise algebraic framework. This approach provides clear geometric intuition and superior computational efficiency without requiring matrices. Moreover, it facilitates tool design, process simulation, and the setup of cutting and grinding operations in a more streamlined and consistent manner. The proposed methodology is illustrated through several numerical examples. Finally, a novel measurement approach is introduced to determine the geometric angles, and an actual measurement is performed to validate the proposed theoretical methods.</p>