"This paper introduces a novel recursive framework for approximating circular geometry and waveforms using discrete segment rotations. Traditional analytic methods, such as the classical circumference formula $C=2pi r$, rely on continuous functions that abstract away the geometric essence of rotation and introduce computational inefficiencies in discrete digital environments. By re-evaluating the 'Method of Exhaustion,' this work derives an original Discrete Radius Formula ($r = frac{C}{2n sin(Deltatheta/2)}$) that eliminates the inherent path-drift found in standard step-based systems. A recursive update algorithm is developed to reconstruct complex signals with $O(1)$ computational complexity, transforming global trigonometric evaluations into local iterative additions. Numerical validation demonstrates high-precision convergence to continuous limits, achieving an absolute error of approximately $7.97 times 10^{-9}$ at high resolution. The results establish a robust bridge between classical geometry and modern digital implementation, offering significant improvements in speed and accuracy for robotics, AI graphics, and signal processing."