Geometric Derivation and Unfolding of Circle Area

This applet demonstrates the geometric derivation of the circle area formula. It shows how a circle can be divided into multiple equal sectors, which are then rearranged to form an approximate rectangle. As the number of divisions increases, the assembled shape progressively approaches a true rectangle, visually illustrating the limiting process at the heart of calculus. Users can interactively adjust the number of sectors and watch in real time as the unfolding pattern becomes increasingly rectangular. This dynamic visualization helps students connect the familiar formula A equals pi r squared to its geometric origins, deepening their conceptual understanding beyond rote memorization. The applet also highlights the relationship between the circle radius, the circumference, and the dimensions of the resulting rectangle: the rectangle length approximates half the circumference, while its height equals the radius. This is an excellent teaching tool for introducing fundamental calculus ideas such as limits and the method of exhaustion in a visual, hands-on manner.