Archimedes’ Approximation of Pi via Inner/Outer Polygons

This GeoGebra example demonstrates how π can be approximated by constructing inscribed and circumscribed regular polygons in a circle and observing how the results improve as the number of sides increases. The key mathematics includes circle geometry (circumference and area), polygonal approximation to a circle, and a sequence/convergence viewpoint behind why the estimates get tighter with larger n; it also leverages angle subdivision closely tied to trigonometric notions. Construction-wise, the model uses commands such as Intersect to locate polygon vertices, Sequence to generate angle/point lists for varying n, Polygon and Circle to build the circle and regular polygons, Segment/Line to form the boundaries, and Perimeter to compute inner and outer perimeter estimates; Area is used to compare inner/outer regions, while Append/Element/Zip help manage iterative objects and display intermediate results (e.g., inside vs. outside perimeter/area). Suitable for high school advanced geometry through early university, especially for teaching π approximation and the intuition of convergence with increasing refinement.