We explore the practical application of isoperimetric inequality (L² ≥ 4πA) to classical methods of circle measurement. Exampling Archimedes' n-gon approach, we compare it to a real-world kinematic scenario of a unit diameter circle rolling on a flat plane surface. Using annular geometry, we demonstrate that π can be derived algebraically by solving for the linear distance its centre travels per full revolution. Unlike exhaustive methods involving non-circular figures, our annular approach begins with isoperimetric equality by deriving a right triangle (with π its lone unknown side) & applying the Pythagorean theorem to it. This algebraic approach to π reveals unexpected yet significant connections between it and the golden ratio. We further explore more assumptions underlying 3.14159... discovering its embedment in an unbounded plane to be catastrophic & remedy with a bounded one. Finally, we close with a fresh new perspective on the notoriously unsolved Riemann Hypothesis problem. Our result suggests both a need for physical experimentation, as well as a need to re-evaluate the general reliability of non-circular methods in rigorously bounding and/or converging on the circle constant π.