Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Tipo de documento: | Dissertação |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFS |
Texto Completo: | http://ri.ufs.br/jspui/handle/riufs/11385 |
Resumo: | This paper aims at presenting a study about the concept and calculation of flat figures areas. For such, it was carried out a historical survey about the development of the idea of area, as well as the precise definition of what could be the area of a flat region. In addition, we seek to show demonstrations of the relationships that are used to determine elementary polygons areas (square, rectangle, parallelogram, triangle, diamond and trapezoid) and also of a circle, where we have tried to present a historical survey since the discovery of the number pi to the perimeter and area relationships. We have also included the notion of areas of curvilinear regions areas calculations by means of Differential and Integral Calculus, which is the most suitable method for dealing with flat surfaces areas delimited by curves. In he same way, we discoursed about the Pick Theorem and the Shoelace Theorem, which are not widely used at elementary and highschool, however, they can be taught in basics mathematics education, in order to enrich knowledge about the methods for determining polygons areas. We finished the stud by discussing equidecomponibility between polygons, which involves decomposing a given polygon P into smaller figures, and by means of a rearrangement of these pieces, to obtain a polygon Q. We also present the Bolyai-Gerwien’s Theorem, which deals, in general terms, with equivalence between polygons with the same areas. |
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Santos, Luiz Carlos DantasRamos, Zaqueu Alves2019-06-19T21:58:56Z2019-06-19T21:58:56Z2019-05-17SANTOS, Luiz Carlos Dantas. Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas. 2019. 63 f. Dissertação (Mestrado profissional em Matemática) - Universidade Federal de Sergipe, São Cristóvão, SE, 2019.http://ri.ufs.br/jspui/handle/riufs/11385This paper aims at presenting a study about the concept and calculation of flat figures areas. For such, it was carried out a historical survey about the development of the idea of area, as well as the precise definition of what could be the area of a flat region. In addition, we seek to show demonstrations of the relationships that are used to determine elementary polygons areas (square, rectangle, parallelogram, triangle, diamond and trapezoid) and also of a circle, where we have tried to present a historical survey since the discovery of the number pi to the perimeter and area relationships. We have also included the notion of areas of curvilinear regions areas calculations by means of Differential and Integral Calculus, which is the most suitable method for dealing with flat surfaces areas delimited by curves. In he same way, we discoursed about the Pick Theorem and the Shoelace Theorem, which are not widely used at elementary and highschool, however, they can be taught in basics mathematics education, in order to enrich knowledge about the methods for determining polygons areas. We finished the stud by discussing equidecomponibility between polygons, which involves decomposing a given polygon P into smaller figures, and by means of a rearrangement of these pieces, to obtain a polygon Q. We also present the Bolyai-Gerwien’s Theorem, which deals, in general terms, with equivalence between polygons with the same areas.Este trabalho tem por objetivo apresentar um estudo realizado sobre o conceito e o cálculo de áreas de figuras planas. Para tal foi realizado um levantamento histórico acerca do desenvolvimento da ideia de área, bem como a definição precisa do que vem a ser a área de uma região plana. Além disso, procuramos expor as demonstrações das relações que são utilizadas para determinar áreas dos polígonos elementares (quadrado, retângulo, paralelogramo, triângulo, losango e trapézio) e também do círculo, onde buscamos apresentar uma sondagem histórica desde a descoberta do número pi, até as relações de perímetro e área. Incluímos também a noção do cálculo de áreas de regiões curvilíneas por meio do Cálculo Diferencial e Integral, o qual é o método mais indicado para tratar sobre áreas de superfícies planas delimitadas por curvas. Do mesmo modo, dissertamos acerca do Teorema de Pick e do Teorema do Cadarço (Shoelace Theorem), os quais não são muito utilizados de forma genérica nos níveis fundamental e médio, contudo podem ser ministrados no ensino básico de matemática, de forma a enriquecer os conhecimentos sobre métodos para determinar áreas de polígonos. Finalizamos o estudo discorrendo sobre equidecomponibilidade entre polígonos, que versa sobre decompor um dado polígono P em figuras menores e por meio de um rearranjo dessas peças, obter um outro polígono Q e também apresentamos o Teorema de Bolyai-Gerwien, que trata, em linhas gerais, sobre equivalência entre polígonos que apresentam áreas iguais.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESSão Cristóvão, SEporMatemáticaSuperfíciesPolígonosDimensõesCurvasÁreaFiguras planasEquivalênciaAreaFlat regionsPolygonsEquivalenceCIENCIAS EXATAS E DA TERRA::MATEMATICAUm breve estudo sobre o conceito e o cálculo de áreas de figuras planasinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisMestrado Profissional em MatemáticaUniversidade Federal de Sergipereponame:Repositório Institucional da UFSinstname:Universidade Federal de Sergipe (UFS)instacron:UFSinfo:eu-repo/semantics/openAccessTEXTLUIZ_CARLOS_DANTAS_SANTOS.pdf.txtLUIZ_CARLOS_DANTAS_SANTOS.pdf.txtExtracted texttext/plain108532https://ri.ufs.br/jspui/bitstream/riufs/11385/3/LUIZ_CARLOS_DANTAS_SANTOS.pdf.txt7fcf62a111891cd2a601031470e5a9c2MD53THUMBNAILLUIZ_CARLOS_DANTAS_SANTOS.pdf.jpgLUIZ_CARLOS_DANTAS_SANTOS.pdf.jpgGenerated Thumbnailimage/jpeg1469https://ri.ufs.br/jspui/bitstream/riufs/11385/4/LUIZ_CARLOS_DANTAS_SANTOS.pdf.jpg051ec36a77856113775ed5d44785917cMD54LICENSElicense.txtlicense.txttext/plain; charset=utf-81475https://ri.ufs.br/jspui/bitstream/riufs/11385/1/license.txt098cbbf65c2c15e1fb2e49c5d306a44cMD51ORIGINALLUIZ_CARLOS_DANTAS_SANTOS.pdfLUIZ_CARLOS_DANTAS_SANTOS.pdfapplication/pdf1016128https://ri.ufs.br/jspui/bitstream/riufs/11385/2/LUIZ_CARLOS_DANTAS_SANTOS.pdfa0acb9563622a5c59cd5c60bbb36adf4MD52riufs/113852019-06-19 18:58:56.895oai:ufs.br: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Repositório InstitucionalPUBhttps://ri.ufs.br/oai/requestrepositorio@academico.ufs.bropendoar:2019-06-19T21:58:56Repositório Institucional da UFS - Universidade Federal de Sergipe (UFS)false |
dc.title.pt_BR.fl_str_mv |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas |
title |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas |
spellingShingle |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas Santos, Luiz Carlos Dantas Matemática Superfícies Polígonos Dimensões Curvas Área Figuras planas Equivalência Area Flat regions Polygons Equivalence CIENCIAS EXATAS E DA TERRA::MATEMATICA |
title_short |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas |
title_full |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas |
title_fullStr |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas |
title_full_unstemmed |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas |
title_sort |
Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas |
author |
Santos, Luiz Carlos Dantas |
author_facet |
Santos, Luiz Carlos Dantas |
author_role |
author |
dc.contributor.author.fl_str_mv |
Santos, Luiz Carlos Dantas |
dc.contributor.advisor1.fl_str_mv |
Ramos, Zaqueu Alves |
contributor_str_mv |
Ramos, Zaqueu Alves |
dc.subject.por.fl_str_mv |
Matemática Superfícies Polígonos Dimensões Curvas Área Figuras planas Equivalência |
topic |
Matemática Superfícies Polígonos Dimensões Curvas Área Figuras planas Equivalência Area Flat regions Polygons Equivalence CIENCIAS EXATAS E DA TERRA::MATEMATICA |
dc.subject.eng.fl_str_mv |
Area Flat regions Polygons Equivalence |
dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::MATEMATICA |
description |
This paper aims at presenting a study about the concept and calculation of flat figures areas. For such, it was carried out a historical survey about the development of the idea of area, as well as the precise definition of what could be the area of a flat region. In addition, we seek to show demonstrations of the relationships that are used to determine elementary polygons areas (square, rectangle, parallelogram, triangle, diamond and trapezoid) and also of a circle, where we have tried to present a historical survey since the discovery of the number pi to the perimeter and area relationships. We have also included the notion of areas of curvilinear regions areas calculations by means of Differential and Integral Calculus, which is the most suitable method for dealing with flat surfaces areas delimited by curves. In he same way, we discoursed about the Pick Theorem and the Shoelace Theorem, which are not widely used at elementary and highschool, however, they can be taught in basics mathematics education, in order to enrich knowledge about the methods for determining polygons areas. We finished the stud by discussing equidecomponibility between polygons, which involves decomposing a given polygon P into smaller figures, and by means of a rearrangement of these pieces, to obtain a polygon Q. We also present the Bolyai-Gerwien’s Theorem, which deals, in general terms, with equivalence between polygons with the same areas. |
publishDate |
2019 |
dc.date.accessioned.fl_str_mv |
2019-06-19T21:58:56Z |
dc.date.available.fl_str_mv |
2019-06-19T21:58:56Z |
dc.date.issued.fl_str_mv |
2019-05-17 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
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masterThesis |
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publishedVersion |
dc.identifier.citation.fl_str_mv |
SANTOS, Luiz Carlos Dantas. Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas. 2019. 63 f. Dissertação (Mestrado profissional em Matemática) - Universidade Federal de Sergipe, São Cristóvão, SE, 2019. |
dc.identifier.uri.fl_str_mv |
http://ri.ufs.br/jspui/handle/riufs/11385 |
identifier_str_mv |
SANTOS, Luiz Carlos Dantas. Um breve estudo sobre o conceito e o cálculo de áreas de figuras planas. 2019. 63 f. Dissertação (Mestrado profissional em Matemática) - Universidade Federal de Sergipe, São Cristóvão, SE, 2019. |
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Mestrado Profissional em Matemática |
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Universidade Federal de Sergipe |
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