A aplicação da razão áurea na resolução de questões matemáticas que envolvem pentágonos e pentagramas.

Abstract

This study investigates the applicability of the golden ratio in solving geometric problems involving pentagons and pentagrams, focusing on its didactic and mathematical efficiency in educational contexts. The golden ratio, represented by the constant ϕ, has historical significance and frequently appears in geometric structures that draw both theoretical and pedagogical interest. This research adopts a quantitative and documentary approach based on the analysis of official exam questions. The selected questions were classified according to predominant geometric content and compared in terms of traditional solution methods versus those that explicitly apply the golden ratio. Three indicators were used for comparative analysis: number of steps, operational complexity, and dependence on the constant ϕ. The results show that in five out of six analyzed questions, the method involving the golden ratio proved more efficient, requiring fewer steps and less algebraic complexity. These findings suggest that the golden ratio may help reduce students’ cognitive load and facilitate more accessible geometric reasoning. The study supports the integration of the golden ratio as a didactic tool in geometry teaching, promoting not only the learning of fundamental mathematical concepts but also the construction of interdisciplinary connections with art, nature, and culture.