Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas

Abstract

This paper aims at presenting a study about the concept and calculation of flat figures areas. For such, it was carried out a historical survey about the development of the idea of area, as well as the precise definition of what could be the area of a flat region. In addition, we seek to show demonstrations of the relationships that are used to determine elementary polygons areas (square, rectangle, parallelogram, triangle, diamond and trapezoid) and also of a circle, where we have tried to present a historical survey since the discovery of the number pi to the perimeter and area relationships. We have also included the notion of areas of curvilinear regions areas calculations by means of Differential and Integral Calculus, which is the most suitable method for dealing with flat surfaces areas delimited by curves. In he same way, we discoursed about the Pick Theorem and the Shoelace Theorem, which are not widely used at elementary and highschool, however, they can be taught in basics mathematics education, in order to enrich knowledge about the methods for determining polygons areas. We finished the stud by discussing equidecomponibility between polygons, which involves decomposing a given polygon P into smaller figures, and by means of a rearrangement of these pieces, to obtain a polygon Q. We also present the Bolyai-Gerwien’s Theorem, which deals, in general terms, with equivalence between polygons with the same areas.