Pappus-Guldin theorems applied to the study of solid modeling with GeoGebra software

Abstract

In this work, we use Geogebra software to simulate the shape of objects (solids) in three dimensions from their photo and real dimensions using spline interpolation. With the reconstructed object, we analyze its volume and surface area using the Pappus-Guldin Theorems (PGT), the theorems that use mathematical analysis ideas to describe the volume and surface area by the sectional area and by the contour curve of the object. In the simulations, we tested the verification of the modeling for known solids (sphere and torus) and then analyzed some objects used in the industry, such as the packaging of products, pet bottles, yogurt containers, coffee powder packaging, aluminum soda cans, and the packaging of chocolate powder. We also analyzed some objects created by rotating bodies, such as the shape of a jar and an aluminum barrel, and also shapes found in nature, such as the shape of a pear and an egg. Modeling allows us to better understand the packaging used in the industry to minimize manufacturing costs and maximize its utility. Thus, we can modify these packages to obtain the best development of how these products are presented to the public, optimizing its format by analyzing its surface and its volume.