Arc Length Approximation: Parabola Locus and Polyline Path Length

This GeoGebra example demonstrates how to approximate the length of a parabola arc using a polyline (polygonal path) made of multiple segments, and how the estimated path length changes as the partition is refined. Key ideas include geometric path length, the parabola arc and its locus/curve, the computation of polyline lengths by measuring successive segments, and the underlying limit concept behind segment refinement (approaching the true arc length as the pieces get smaller). Construction highlights: GeoGebra uses PolyLine to build the piecewise linear path (e.g., from constructed points into objects like pol1/pol), and then applies Length to measure the total polyline length (yielding lon or its associated length value). The setup also includes curve objects (such as functions f and g) and auxiliary construction points/parameters (e.g., L, c1, M, reg) to place sampling points along the parabola so the arc and its polygonal trace can be compared. Suitable for advanced middle school through early university use, especially for visualizing curve-length concepts and introducing numerical approximation via geometric refinement.