Geometric Approximation of Pi: Area and (Definite) Integral Thinking

This example demonstrates how to approximate π by comparing geometric areas that form upper and lower bounds around a circle. The key mathematical ideas are area representation (linked to definite integrals), constructing polygonal approximations, and analyzing the difference and slope to observe how the approximation error changes as the partition becomes finer. It also supports the use of simple logical checks to update which bound is tighter as parameters vary. Construction focuses on creating circle geometry, generating inscribed/circumscribed polygon (or partition-based) approximations, measuring areas, computing the area difference (circle area minus/plus the approximation), and using those quantities to visualize convergence/accuracy trends. Suitable for advanced secondary school and early calculus courses, especially for building intuition between geometric area, definite integrals, and limiting behavior.