Áreas e perímetros do círculo utilizando aproximações: uma aplicação do método da exaustão

Abstract

This work aims to present the Exhaustion Method as the transition between classical geometry and the modern concept of limit. It develops a historical and mathe matical analysis, highlighting the contributions of Eudoxus of Cnidus and Archimedes of Syracuse, who used this method to determine the areas and perimeters of curved figures, approximating the value of the number π. By constructing regular polygons inscribed and circumscribed within a circle, it is demonstrated that, as the number of sides increases, their areas and perimeters tend, respectively, to the values of π and 2π. This approach demonstrates the idea of successive approximation and the in tuitive origin of the concept of limit. It is concluded that the Method of Exhaustion represents an important link between antique geometric thought and the development of infinitesimal calculus, in addition to constituting a relevant teaching tool for understan ding the fundamentals of geometry and mathematical analysis.