Center of Gravity of Polygonal Regions and Conic Sections

Abstract

In this paper, we will consider the center of gravity, or the center of mass, of several types of polygonal regions in the XY-plane. In order to calculate the center of gravity of such regions, instead of techniques from calculus, one can conveniently use geometric ideas. Our regions are of variable nature, so one can consider the locus of their center of gravity in the XY-plane. The locus of the center of gravity of one region we have considered turns out to be a hyperbola with axes parallel to the coordinate axes. This observation can be reversed to come up with a new definition for a hyperbola without involving eccentricity. We will use this new definition along with the dynamic geometry software Geometer's Sketchpad to present a novel construction of a hyperbola. The locus of the center of gravity of other polygonal regions we have considered yield some other interesting curves as well. In addition to Geometer's Sketchpad, we have also used the computer algebra system Mathematica to facilitate the center of gravity calculations. The paper amply demonstrates the usage of different types of software hand-in-hand to experiment and conjecture mathematical results.