Geometric Limit Sequence of Inscribed Circles and Nested Squares

This applet demonstrates the geometric construction and limit process involving a square, its inscribed circles, and nested squares. Starting with an outer square, the applet constructs its inscribed circle, then an inscribed square within that circle, and continues this iterative process indefinitely. At each step, the new square is rotated 45 degrees relative to the previous one, with its vertices touching the midpoints of the sides of the preceding square. The applet uses GeoGebra commands such as Rotate, Midpoint, Line Segment, and Sequence to dynamically visualize each stage of the construction. As the number of iterations increases, students can observe how the areas and side lengths of the nested shapes progressively shrink. The Sequence feature displays multiple stages simultaneously, allowing learners to see the pattern emerge clearly. This visual approach helps students grasp the concept of geometric convergence and understand how a sequence of nested figures approaches a limiting point. The ratio between consecutive side lengths is a constant factor of 1 over the square root of 2, forming a geometric sequence whose sum converges to a finite value. This applet is particularly suitable for introductory calculus courses covering limits and geometric series, providing an intuitive bridge between abstract algebraic formulas and concrete geometric reasoning. Students can manipulate the iteration count and explore how the sum of areas relates to the area of the original square, reinforcing the connection between infinite processes and finite results.