2025-10-302025-10-302025-07-11https://repositorio.ifal.edu.br/handle/123456789/1299This work aims to investigate the geometric and analytical properties of the cycloid family, including variations such as the common cycloid, curtate cycloid, and prolate cycloid. The research is conducted through a comparative study that covers both theoretical and practical aspects of these curves, analyzing their parametric equations, derivatives, integrals, and geometric characteristics. Additionally, the study explores historical and contemporary applications of cycloids in contexts such as physics, engineering, and applied mathematics, highlighting classical problems such as the brachistochrone and the tautochrone. The work seeks to contribute to a deeper understanding of these curves from both mathematical and educational perspectives, serving as a resource for teaching differential and integral calculus.ptAttribution-NonCommercial 3.0 BrazilMatemáticaCicloidesGeometria diferencialCurvas paramétricasCálculo diferencialBraquistócronoCycloidGeometryParametric curvesBrachistochroneDifferential calculusTrabalho de Conclusão de CursoCIENCIAS EXATAS E DA TERRA::MATEMATICA