Approximating Pi with Inscribed and Circumscribed Polygons

This GeoGebra example demonstrates how to approximate the mathematical constant π by using regular polygons inscribed in and circumscribed around a circle. The core concept is the limit process: as the number of sides n increases, the perimeter of the inscribed polygon approaches the circle's circumference from below, while the circumscribed polygon's perimeter approaches from above, yielding upper and lower bounds for π. The construction uses GeoGebra commands such as Polygon to create the polygons, Circle for the unit circle, Textfield and InputBox for interactive side-number input, Zip and list operations to compute perimeters, and segment objects (seg1, seg2) to visualize side lengths. Text objects (text1, text2) display the current approximation values in real time. This is ideal for high school or early college students exploring limits, the nature of π, or the historical method of exhaustion (Liu Hui's circle cutting). It also serves as an excellent interactive tool for mathematics labs and inquiry-based learning.