Sobre o cálculo de áreas e o teorema de Pick

Abstract

This work aims to introduce an alternative proposal for the teaching of the concept and calculation of the area of several polygons. Before, we strove to try showing the importance of the Geometry teaching and of the concept of area, and we tried to analyze, in the light of PAVANELLO, a Brazilian education phenomenon: the abandonment of Geometry teaching. After, we follow with the definitions and basic concepts required to the treatment of the concept and calculation of the more elementary polygons area as squares, rectangles, triangles, etc. On third chapter, we showed some results that can be demonstrated with the propositions of the previous chapter, as the Pythagoras’s Theorem, The Law of Sines and the Fundamental trigonometric limit. After we introduce the Pick’s Theorem, its versions for the polygons with "holes" and for polygons with vertices of reasonable coordinates and its surprising equivalence with the Euler’s Formula for polygons in plan. On the fifth chapter, we show some interesting results that can be obtained by using the Pick’s Theorem, like, for instance, an estimate for the value of π, and find a entire solution for the diophantine equation mp−nq = 1, where p e q are coprime. Finally, on the sixth chapter, we introduce an educational proposal for the treatment of the concept of polygon’s area, that makes use of the Google Maps site and of the Geogebra Software, together with the Pick’s Theorem and of the relation between areas of similar figures. At this activity, we propose to the students calculate the approximated area of regions (as the State of Alagoas, the city of Maceio) just using the results presented at classroom, under the supervision of the teacher. At the end of work, it follows an appendix with a small abstract about the integrals theory and its relation with the calculation of area under the graphic of functions and the definition of the logarithmic function.