O princípio de Cavalieri para cálculo de volumes no ensino médio: algumas possibilidades

Abstract

In this work we present a sequence of activities using the Cavalieri’s Principle in calculating volumes of geometric solids. These activities are aimed at high school students who know how to calculate the volume of a rectangular block. From this condition, we introduce Cavalieri’s Principle as an axiom in activities that awaken students’ intuition so that they understand and accept this principle. Students are led to participate actively in the derivation of formulas for calculating the volume of solids as traditional pyramids, cones, cylinders and spheres. With the help of Dynamic Geometry software GeoGebra 5.0 Beta JOGL1 3D, we create pictures and animations that will facilitate the visualization of geometric solids as well as the creation of conjectures as assist students in understanding formulas. Besides these solids which will we call traditional, other examples will be presented in order to apply the Cavalieri’s Principle which are not commonly found in books aimed at this level of education: the torus, a sphere with a cylindrical bore. As a final activity, we consider the solid formed by the intersection by two cylinders of the same radius with their perpendicular axes, their heights are greater than the cylinder diameter of the base. This volume calculation is commonly presented in College textbooks using the Integral Calculus, but in this work we take students to understand the formula by calculating the volume using the Cavalieri’s Principle.