{"id":1136,"date":"2020-02-06T19:58:51","date_gmt":"2020-02-06T22:58:51","guid":{"rendered":"https:\/\/www.blogs.unicamp.br\/zero\/?p=1136"},"modified":"2023-08-25T12:17:09","modified_gmt":"2023-08-25T15:17:09","slug":"quantos-graus-tem-o-angulo-interno-de-um-poligono-regular-de-infinitos-lados","status":"publish","type":"post","link":"https:\/\/www.blogs.unicamp.br\/zero\/1136\/","title":{"rendered":"Quantos graus tem o \u00e2ngulo interno de um pol\u00edgono regular de infinitos lados?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"1136\" class=\"elementor elementor-1136\" data-elementor-settings=\"{&quot;ha_cmc_init_switcher&quot;:&quot;no&quot;}\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0a401e5 elementor-section-boxed elementor-section-height-default elementor-section-height-default jltma-glass-effect-no\" data-id=\"0a401e5\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8e5e265 jltma-glass-effect-no\" data-id=\"8e5e265\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5c3489b jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"5c3489b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/www.blogs.unicamp.br\/zero\/2020\/02\/06\/how-many-degrees-does-the-internal-angle-of-a-regular-polygon-with-infinite-sides\/\">(Translate)<\/a><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1e0ed68 elementor-section-boxed elementor-section-height-default elementor-section-height-default jltma-glass-effect-no\" data-id=\"1e0ed68\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2223d30 jltma-glass-effect-no\" data-id=\"2223d30\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c54faf0 jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"c54faf0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">Quem nunca se viu questionando quantos graus deveria ter cada \u00e2ngulo interno de um pol\u00edgono regular de infinitos lados? Mais popularmente chamado de infinit\u00f3gono regular. Essa deve ser uma pergunta recorrente na maioria das pessoas (pelo menos naquelas com tempo e paci\u00eancia suficientes para ler este blog).<\/p><p align=\"justify\">Para analisar este problema, partiremos do fato que a soma dos \u00e2ngulos internos de um tri\u00e2ngulo \u00e9 180 graus. Uma matem\u00e1tica bastante simples, mas que mostraremos ser suficiente para este c\u00e1lculo com um infinit\u00f3gono regular.<\/p><p align=\"justify\">No caso, se o tri\u00e2ngulo regular tem todos os \u00e2ngulos iguais (por defini\u00e7\u00e3o de ser um pol\u00edgono regular), ent\u00e3o cada \u00e2ngulo interno dele deve ter 180\/3 graus. Ou seja, 60 graus.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1f9852e jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"1f9852e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"260\" height=\"256\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/triangulo-black.png\" class=\"attachment-large size-large wp-image-1807\" alt=\"\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/triangulo-black.png 260w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/triangulo-black-24x24.png 24w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/triangulo-black-48x48.png 48w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/triangulo-black-96x96.png 96w\" sizes=\"(max-width: 260px) 100vw, 260px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-07a6842 jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"07a6842\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">No caso de um quadrado (um pol\u00edgono regular de 4 lados). Podemos dividir um quadrado a partir de seu centro e criar assim 4 tri\u00e2ngulos. A soma dos \u00e2ngulos do seu centro devem dar ao todo 360 graus, pois se completam. Dividindo 360 graus pelo n\u00famero de lados, no caso 4, chegamos que a soma dos outros dois \u00e2ngulos deve ser igual ao \u00e2ngulo interno do pol\u00edgono, no caso, 45+45 graus, ou seja, 90 graus.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5f9017c jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"5f9017c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"297\" height=\"295\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/quadrado-black.png\" class=\"attachment-large size-large wp-image-1808\" alt=\"\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/quadrado-black.png 297w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/quadrado-black-150x150.png 150w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/quadrado-black-24x24.png 24w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/quadrado-black-48x48.png 48w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/quadrado-black-96x96.png 96w\" sizes=\"(max-width: 297px) 100vw, 297px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-817391f jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"817391f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">No caso de um pent\u00e1gono regular (um pol\u00edgono regular de 5 lados). Podemos dividi-lo a partir de seu centro e criar assim 5 tri\u00e2ngulos. A soma dos \u00e2ngulos do seu centro devem dar ao todo 360 graus, pois se completam. Dividindo 360 graus pelo n\u00famero de lados, no caso 5, chegamos que a soma dos outros dois \u00e2ngulos deve ser igual ao \u00e2ngulo interno do pol\u00edgono, no caso, 54+54 graus, ou seja, 108 graus.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e35128b jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"e35128b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"251\" height=\"251\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/pent\u00e1gono-black.png\" class=\"attachment-large size-large wp-image-1809\" alt=\"\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/pent\u00e1gono-black.png 251w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/pent\u00e1gono-black-150x150.png 150w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/pent\u00e1gono-black-24x24.png 24w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/pent\u00e1gono-black-48x48.png 48w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2020\/02\/pent\u00e1gono-black-96x96.png 96w\" sizes=\"(max-width: 251px) 100vw, 251px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2e316dc jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"2e316dc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">Para o quadrado a resposta poderia ser intuitiva, mas o racioc\u00ednio serve tanto para o pent\u00e1gono (como mostramos acima) como para figuras de mais lados.<\/p>\n<p align=\"center\"><i>6 lados: [180 \u2013 (360\/6)] = 120 graus<br>7 lados: [180 \u2013 (360\/7)] = 128,5 graus<br>8 lados: [180 \u2013 (360\/8)] = 135 graus<br>9 lados: [180 \u2013 (360\/9)] = 140 graus<br>10 lados: [180 \u2013 (360\/10)] = 144 graus<br>\u2026<br>100 lados: [180 \u2013 (360\/100)] = 176,4 graus<br>\u2026<br>1.000 lados: [180 \u2013 (360\/1.000)] = 179,6 graus<br>\u2026<br>1.000.000 lados: [180 \u2013 (360\/1.000.000)] = 179,9996 graus<br>\u2026<br>1.000.000.000 lados: [180 \u2013 (360\/1.000.000.000)] = 179,9999996 graus<\/i><\/p>\n<p align=\"justify\">Aplicando um limite simples \u00e0 fun\u00e7\u00e3o graus desse pol\u00edgono regular, podemos chegar na seguinte express\u00e3o:<\/p>\n<p align=\"center\"><i>180 \u2013 lim (360\/x) = 180 \u2013 0 = 180.<sup><br>x\u2192\u221e &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<\/sup> <\/i><\/p>\n<p align=\"justify\">Ou seja, quando o n\u00famero de lados tender ao infinito, a medida de cada \u00e2ngulo interno do pol\u00edgono \u00e9 180 graus. Incr\u00edvel pensar como ele se fecha apenas com \u00e2ngulos rasos.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8b0f833 elementor-section-boxed elementor-section-height-default elementor-section-height-default jltma-glass-effect-no\" data-id=\"8b0f833\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1e3e88f jltma-glass-effect-no\" data-id=\"1e3e88f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b6c68cd jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"b6c68cd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/www.blogs.unicamp.br\/zero\">Voltar para p\u00e1gina principal<\/a><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0e8a2cf elementor-section-boxed elementor-section-height-default elementor-section-height-default jltma-glass-effect-no\" data-id=\"0e8a2cf\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d0562ea jltma-glass-effect-no\" data-id=\"d0562ea\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-47a7a46 jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"47a7a46\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/www.blogs.unicamp.br\/zero\/quem-escreve-os-posts\/\">Quem escreve os posts?<\/a><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Um pol\u00edgono regular de infinitos lados apenas parece um c\u00edrculo.<\/p>\n","protected":false},"author":434,"featured_media":1143,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"colormag_page_container_layout":"default_layout","colormag_page_sidebar_layout":"default_layout","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"pgc_sgb_lightbox_settings":"","_vp_format_video_url":"","_vp_image_focal_point":[],"footnotes":""},"categories":[1210],"tags":[],"class_list":["post-1136","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-v-3-ed-1"],"_links":{"self":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/1136","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/users\/434"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/comments?post=1136"}],"version-history":[{"count":16,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/1136\/revisions"}],"predecessor-version":[{"id":5219,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/1136\/revisions\/5219"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/media\/1143"}],"wp:attachment":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/media?parent=1136"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/categories?post=1136"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/tags?post=1136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}