Calculating the volume of convex polyhedra using GeoGebra 3D
DOI:
https://doi.org/10.23925/2237-9657.2026.v15i1p133-149Palabras clave:
icosidodecahedron, classes of polyhedra, nested radicalsResumen
This article presents the decomposition of the icosidodecahedron, an Archimedean convex polyhedron that is the dual of the rhombic triacontahedron, a Catalan polyhedron. The decomposition into regular triangular and pentagonal pyramids, performed in GeoGebra 3D, aims to establish a strategy for calculating the polyhedron's volume. In the polyhedron construction stage, which precedes the decomposition stage, we imported information, such as the coordinates of the vertices and faces, from the Visual Polyhedra Platform. In the calculation of the polyhedron's volume, we converted nested radicals into sums and differences of simple radicals. We made the decomposition available on a GeoGebra platform page accessible via an external link and illustrated the process with polyhedra of other classes. The decomposition strategy proved effective in calculating the volume, which agrees with data available in the literature.
Descargas
Citas
Abar, C. A. A. P & Alencar, S. V. (2013). The Instrumental Genesis and its interaction with GeoGebra: a proposal for continuing education for mathematics teachers. Bolema, 27(46), 349–365.
Catalan, M. E. (1865). Memoire sur la theorie des polyedres. Journal de l'ecole Imperiale Polytechnique, 24(41), 1–71.
Coxeter, H. S. M. (1973). Regular polytopes. New York: Dover.
Cromwell, P. R. (2008). Polyhedra. Cambridge: Cambridge University Press.
GeoGebra. (2026). GeoGebra manual. Available at: https://geogebra.github.io/docs/reference/en/GeoGebra_Installation/. Accessed on: 28/12/2025.
Hart, G. W. (2025). Encyclopedia of polyhedra. Available at: https://geometrycode.com/encyclopedia-of-polyhedra-by-george-w-hart-and-other-geometric-gems/. Accessed on: 06/04/2026.
Johnson, N. W. (1966). Convex polyhedra with regular faces. Canadian Journal of Mathematics, 18, 169–200.
Landau, S. (1994). How to tangle with a nested radical. The Mathematical Intelligencer, 16, 49–55.
Mathias, C. V., Silva, C. M. da & Simas, F. L. B. (2024). Spatial visualization skills present in items of the Brazilian high school national exam. EURASIA Journal of Mathematics, Science and Technology Education, 20(3), 1–11.
McCooey, D. I. (2015a). Visual polyhedra. Available at: https://dmccooey.com/polyhedra/. Accessed on: 28/12/2025.
McCooey, D. I. (2015b). Icosidodecahedron. Available at: https://dmccooey.com/polyhedra/Icosidodecahedron.html. Accessed on: 04/04/2026.
McCooey, D. I. (2015c). Icosidodecahedron coordinates. Available at: https://dmccooey.com/polyhedra/Icosidodecahedron.txt. Accessed on: 04/04/2026.
McCooey, D. I. (2015d). Rhombic triacontahedron. Available at: https://dmccooey.com/polyhedra/RhombicTriacontahedron.html. Accessed on: 04/04/2026.
McCooey, D. I. (2015e). Triangular hebesphenorotunda. Available at: https://dmccooey.com/polyhedra/TriangularHebesphenorotunda.html. Accessed on: 04/04/2026.
Mindat. (2026a). Spessartine. Available at: https://www.mindat.org/min-3725.html. Accessed on: 04/04/2026.
Mindat. (2026b). Grossular. Available at: https://www.mindat.org/min-1755.html. Accessed on: 04/04/2026.
Nós, R. L., Saito, O. H. & Santos, M. A. dos. (2017a). Geometria, radicais duplos e a raiz quadrada de números complexos. Revista Eletrônica Paulista de Matemática, 11, 48–64. https://doi.org/10.21167/cqdvol11201723169664rlnohsmas4864.
Nós, R. L., Saito, O. H. & Santos, M. A. dos. (2017b). Radicais duplos e a raiz quadrada de um número complexo. In: Proceeding Series of the Brazilian Society of Computational and Applied Mathematics 5(1) (pp. 010557-1– 010557-7). Gramado, RS. https://doi.org/10.5540/03.2017.005.01.0557.
Nós, R. L & Silva, V. M. R. da. (2019a). Radicais duplos no cálculo do volume de poliedros convexos. Revista Eletrônica Paulista de Matemática, 16, 53–70. https://doi.org/10.21167/cqdvol16201923169664rlnvmrs5370.
Nós, R. L. & Silva, V. M. R. da. (2019b). Compondo/decompondo poliedros convexos com o GeoGebra 3D. In: Proceeding Series of the Brazilian Society of Computational and Applied Mathematics 7(1) (pp. 010364-1– 010364-7). Uberlândia, MG. https://doi.org/10.5540/03.2020.007.01.0364.
Pugh, A. (1976). Polyhedra: a visual approach. Los Angeles: University of California Press.
Rahal, S. (2017). Introduction to nested radicals. Stockholm: Stockholms Universitet.
Silva, V. M. R. da (2018). Calculando o volume de poliedros convexos. 2018. 129f. Trabalho de Conclusão de Curso (Licenciatura em Matemática) – Universidade Tecnológica Federal do Paraná, Curitiba, PR. https://repositorio.utfpr.edu.br/jspui/handle/1/9047.
Silva, V. M. R. da & Nós, R. L. (2018). Calculando o volume de poliedros convexos. Curitiba, PR: CRV. https://doi.org/10.24824/978854442681.4.
Silva, V. M. R. da & Nós, R. L. (2022). Using GeoGebra 3D in the composition and decomposition of convex polyhedra for volume calculation. Journal of Engineering Research, 3(2), 1–11. https://doi.org/10.22533/at.ed.3173222221210.
Silva, V. M. R. da, Nós, R. L. & Sano, M. (2023). Uma visão dinâmica do teorema de Pitágoras via GeoGebra. Revista do Instituto GeoGebra Internacional de São Paulo, 12(1), 62–77. https://doi.org/10.23925/2237-9657.2023.v12i1p062-077.
Weisstein, E. W. (2026). Dual polyhedron. Available at: https://mathworld.wolfram.com/DualPolyhedron.html. Accessed on: 05/04/2026.
Descargas
Publicado
Cómo citar
Número
Sección
Licencia
Derechos de autor 2026 Revista do Instituto GeoGebra Internacional de São Paulo. ISSN 2237-9657
Esta obra está bajo una licencia internacional Creative Commons Atribución 4.0.
No se cobrarán tasas de envío, procesamiento y publicación de los trabajos enviados a la revista, ni el registro del DOI en la Crossref.
Los autores mantienen sus derechos de autor y conceden a la revista el derecho a la primera publicación del trabajo, simultáneamente licenciada bajo una licencia Creative Commons - Atribución 4.0 Licencia Internacional CC BY, que permite que otros compartan el trabajo con un reconocimiento de la autoría del mismo y publicación inicial en esta revista.
La revista GeoGebra incentiva a que sus autores registren sus trabajos en sistemas orientados a la gestión de información y comunicación de investigadores, tales como Academia.edu, Mendeley, ResearchGate etc.
10.23925
