Inscribed and Circumscribed Polygons

This applet demonstrates the geometric construction of polygons inscribed in and circumscribed about a circle. Using GeoGebra's Sequence and Circle commands, it dynamically generates both inscribed and circumscribed regular polygons whose number of sides can be adjusted interactively. As the user increases the number of sides, the applet visually shows how the perimeters and areas of both types of polygons progressively approach those of the circle itself. This provides an intuitive, hands-on illustration of the mathematical concept of limits, where the polygon becomes indistinguishable from the circle as the side count grows without bound. The inscribed polygons always lie inside the circle with all vertices touching the circumference, while the circumscribed polygons enclose the circle with each side tangent to it. Students can manipulate the side count slider to observe the convergence in real time, comparing the computed perimeter and area values against the known formulas for the circle. This visual approach reinforces key concepts in geometry and calculus, including the idea that regular polygons can approximate circular measurements with arbitrary precision. The dynamic construction also highlights the underlying geometric principles involved in classical polygon construction, making it a valuable teaching tool for students studying limit processes, geometric approximation, and the relationship between discrete polygonal shapes and continuous circular forms.