{"id":12180,"date":"2023-11-29T10:00:00","date_gmt":"2023-11-29T13:00:00","guid":{"rendered":"https:\/\/www.blogs.unicamp.br\/revista\/?p=12180"},"modified":"2024-05-02T19:50:42","modified_gmt":"2024-05-02T22:50:42","slug":"porque-o-angulo-da-ligacao-tetraedrica-e-109","status":"publish","type":"post","link":"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/porque-o-angulo-da-ligacao-tetraedrica-e-109\/","title":{"rendered":"Porque o \u00e2ngulo da liga\u00e7\u00e3o tetra\u00e9drica \u00e9 109\u00b0?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"12180\" class=\"elementor elementor-12180\" data-elementor-settings=\"{&quot;ha_cmc_init_switcher&quot;:&quot;no&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5126c9d8 e-flex e-con-boxed jltma-glass-effect-no e-con e-parent\" data-id=\"5126c9d8\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t<div 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elementor-widget-heading\" data-id=\"2d792846\" 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<h2 class=\"elementor-heading-title elementor-size-default\">Porque o \u00e2ngulo da liga\u00e7\u00e3o tetra\u00e9drica \u00e9 109\u00b0?<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-31077790 elementor-widget__width-inherit jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"31077790\" 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=\"1201\" height=\"901\" src=\"https:\/\/www.blogs.unicamp.br\/revista\/wp-content\/uploads\/sites\/279\/2023\/11\/Porque_o_a_ngulo_da_ligac_a_o_tetrae_drica_e_109_1.webp\" class=\"attachment-full size-full 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sizes=\"(max-width: 1201px) 100vw, 1201px\" \/>\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-1f3472ef jltma-glass-effect-no elementor-widget elementor-widget-ha-post-info happy-addon ha-post-info\" data-id=\"1f3472ef\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"ha-post-info.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"ha-inline-items ha-icon-list-items ha-post-info\">\n\t\t\t\t\t<li class=\"ha-icon-list-item elementor-repeater-item-0fdbbfb ha-inline-item\" itemprop=\"datePublished\">\n\t\t\t\t\t\t<a href=\"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"ha-icon-list-icon\">\n\t\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-calendar\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M12 192h424c6.6 0 12 5.4 12 12v260c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V204c0-6.6 5.4-12 12-12zm436-44v-36c0-26.5-21.5-48-48-48h-48V12c0-6.6-5.4-12-12-12h-40c-6.6 0-12 5.4-12 12v52H160V12c0-6.6-5.4-12-12-12h-40c-6.6 0-12 5.4-12 12v52H48C21.5 64 0 85.5 0 112v36c0 6.6 5.4 12 12 12h424c6.6 0 12-5.4 12-12z\"><\/path><\/svg>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t<span class=\"ha-icon-list-text ha-post-info__item ha-post-info__item--type-date\">\n\t\t\t\t\t\t\t\t\t\t29\/11\/2023\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t<\/a>\n\t\t\t\t<\/li>\n\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-451c60a4 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"451c60a4\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-5f2ae628 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"5f2ae628\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-1a6037f2 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"1a6037f2\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-3b675f0d jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"3b675f0d\" 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\tComo podemos obter o \u00e2ngulo da liga\u00e7\u00e3o tetra\u00e9drica? Conhe\u00e7a o processo matem\u00e1tico por tr\u00e1s desse intrigante mist\u00e9rio da Qu\u00edmica!\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-ed7ff73 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"ed7ff73\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-50ea849b e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"50ea849b\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-6e8a9982 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"6e8a9982\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-505abf36 jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"505abf36\" 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<h2 class=\"elementor-heading-title elementor-size-default\">Autor\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6b66f8ba jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"6b66f8ba\" 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<p class=\"elementor-heading-title elementor-size-default\">Zero<\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-81a4af9 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"81a4af9\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a7ea5a9 jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"a7ea5a9\" 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><\/p><p>Na minha \u00faltima aula de Qu\u00edmica Geral (j\u00e1 contei que estou cursando Licenciatura em Qu\u00edmica?), vimos sobre \u00e2ngulos de liga\u00e7\u00e3o entre \u00e1tomos e os nomes que essas geometrias recebiam.<\/p><p> <\/p><figure><img decoding=\"async\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2023\/10\/tetraedro-1-1024x561.jpeg\" alt=\"\" \/><\/figure><p> <\/p><p>No caso das geometrias planas (<strong>linear<\/strong> e <strong>trigonal planar<\/strong>), era f\u00e1cil entender porque os \u00e2ngulos de 180\u00b0 e 120\u00b0 apareciam (dividir um c\u00edrculo de 360\u00b0 em 2 ou 3 partes).<\/p><p> <\/p><p>Na geometria <strong>bipiramidal trigonal<\/strong> o racioc\u00ednio segue parecido, basta enxergar um <strong>trigonal planar <\/strong>combinado com um <strong>linear<\/strong>, justificando os 120\u00b0 e 180\u00b0.<\/p><p> <\/p><p>Na geometria <strong>octa\u00e9drica<\/strong> &#8220;vemos&#8221; um monte de \u00e2ngulos retos, ent\u00e3o \u00e9 f\u00e1cil entender porque os \u00e2ngulos de 90\u00b0.<\/p><p> <\/p><p>Contudo, voc\u00ea percebeu que pulamos a geometria <strong>tetra\u00e9drica<\/strong>?<\/p><p> <\/p><p>\u00c9 dito que para ela o \u00e2ngulo de liga\u00e7\u00e3o \u00e9 de 109\u00b0&#8230; mas porque?<\/p><p> <\/p><p>Voc\u00ea pode at\u00e9 justificar dizendo que \u00e9 o \u00e2ngulo que faz o maior afastamento entre os \u00e1tomos&#8230; mas porque 109\u00b0 e n\u00e3o 112\u00b0 ou 106\u00b0?<\/p><p> <\/p><p>\u00a0<\/p><p> <\/p><p>Na minha \u00faltima aula de Pr\u00e9-C\u00e1lculo para professores de Qu\u00edmica (agora digo minha aula, pois era eu que estava ministrando), fazia a revis\u00e3o dos conte\u00fados das aulas anteriores, e ap\u00f3s tratar logaritmos, quis revisar de uma forma mais pr\u00e1tica fun\u00e7\u00f5es trigonom\u00e9tricas. Ent\u00e3o passei esta geometria molecular e prop\u00fas algo divertido para a turma.<\/p><p> <\/p><p><em>Se algu\u00e9m ali, nos pr\u00f3ximos 40 minutos, conseguisse mostrar porque aquele \u00e2ngulo \u00e9 de 109\u00b0, a turma inteira (que assinou a presen\u00e7a naquele dia), ganharia mais 0,5 na m\u00e9dia. Detalhe: valia tudo o que estivesse ao alcance da turma, desde pesquisar na internet, no chat-gpt, procurar professores ou veteranos pra responder, mas eles deveriam ser capazes de &#8220;explicar&#8221; porque s\u00e3o 109\u00b0.<\/em><\/p><p> <\/p><p>(Isso significa que poderiam ser utilizados recursos quaisquer da matem\u00e1tica, desde que quem fosse explicar, conseguisse entender o que est\u00e1 sendo feito. Sendo capaz de justificar todos os procedimentos adotados)<\/p><p> <\/p><p>(Essa proposta colaborativa foi bastante motivadora e penso us\u00e1-la novamente na pr\u00f3xima aula de revis\u00e3o, pois mesmo quem n\u00e3o precisava daquele 0,5 na m\u00e9dia, tinha a inten\u00e7\u00e3o de querer ajudar o restante da turma)<\/p><p> <\/p><p>Quase estourando o tempo, conseguiram encontrar a explica\u00e7\u00e3o e process\u00e1-la de um modo que justificasse o motivo \ud83d\ude42<\/p><p> <\/p><p>\u00a0<\/p><p> <\/p><p>Ficou curioso para saber como encontrar esse \u00e2ngulo?<\/p><p> <\/p><p>Ent\u00e3o mostrarei agora a resolu\u00e7\u00e3o que esperava dos meus alunos (e que fiz para eles ap\u00f3s a explica\u00e7\u00e3o de sua colega).<\/p><p> <\/p><p>Comece imaginando um tetraedro regular:<\/p><p> <\/p><figure><img decoding=\"async\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2023\/10\/image-10.png\" alt=\"\" \/><\/figure><p> <\/p><p>Ent\u00e3o vamos pegar uma de suas faces (por exemplo a base) e calcular o seu ap\u00f3tema (que seria a altura de uma aresta que divide o pol\u00edgono em tri\u00e2ngulos is\u00f3sceles congruentes).<\/p><p> <\/p><figure><img decoding=\"async\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2023\/10\/image-22.png\" alt=\"\" \/><\/figure><p> <\/p><p>Como estamos procurando o \u00e2ngulo, n\u00e3o faz mal escolhermos uma medida para os lados do tri\u00e2ngulo, neste caso vou assumir que o tri\u00e2ngulo tem lado 2. Agora sabendo que o \u00e2ngulo interno do tri\u00e2ngulo equil\u00e1tero \u00e9 de 60\u00b0, e que os tri\u00e2ngulos internos s\u00e3o is\u00f3sceles, ent\u00e3o seu menor \u00e2ngulo ser\u00e1 de 30\u00b0. Logo o ap\u00f3tema A poder\u00e1 ser descrito como:<\/p><p> <\/p><p>tang(30\u00b0) = A\/1 = A<\/p><p> <\/p><p>Ent\u00e3o vou calcular a altura de uma das faces do tetraedro:<\/p><p> <\/p><figure><img decoding=\"async\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2023\/10\/image-23.png\" alt=\"\" \/><\/figure><p> <\/p><p>Sabendo que seu \u00e2ngulo interno \u00e9 60\u00b0, ent\u00e3o sua altura H ser\u00e1:<\/p><p> <\/p><p>tang(60\u00b0) = H\/1 = H<\/p><p> <\/p><p>Por fim, vou procurar descobrir o valor da altura X do tetraedro, utilizando o teorema de pit\u00e1goras nesse tri\u00e2ngulo imagin\u00e1rio que criei, onde A \u00e9 um dos catetos e H \u00e9 a hipotenusa:<\/p><p> <\/p><figure><img decoding=\"async\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2023\/10\/image-15.png\" alt=\"\" \/><\/figure><p> <\/p><p>Temos que:<\/p><p> <\/p><p>Tang(30\u00b0)\u00b2 + X\u00b2 = Tang(60\u00b0)\u00b2<\/p><p> <\/p><p>Substituindo os valores da tangente de 30\u00b0 e 60\u00b0, fica:<\/p><p> <\/p><p>(\u221a3\/3)\u00b2 + X\u00b2 + \u221a3\u00b2<\/p><p> <\/p><p>X\u00b2 = 3 &#8211; 3\/9<\/p><p> <\/p><p>X\u00b2 = 24\/9<\/p><p> <\/p><p>X = \u221a(24\/9) = 2\u221a6\/3<\/p><p> <\/p><p>Assim, sabendo que a altura do tetraedro \u00e9 3\/2 e sua aresta vale 2, podemos expressar o valor do \u00e2ngulo \u03b8, como:<\/p><p> <\/p><figure><img decoding=\"async\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2023\/10\/image-19.png\" alt=\"\" \/><\/figure><p> <\/p><p>cos(\u03b8) = (2\u221a6\/3)\/2 = \u221a6\/3<\/p><p> <\/p><p>\u03b8 = arccos(\u221a6\/3) ~ 35.26\u00b0<\/p><p> <\/p><p>Retomando ao nosso problema, estamos procurando o \u00e2ngulo de liga\u00e7\u00e3o \u03b1 que vai do centro do tetraedro at\u00e9 seus v\u00e9rtices. Assim, essas liga\u00e7\u00f5es formar\u00e3o tri\u00e2ngulos is\u00f3sceles, logo dois dos seus \u00e2ngulos ser\u00e3o iguais. Mas j\u00e1 descobrimos que \u03b8 = arccos(\u221a6\/3), com isso basta fazer a diferen\u00e7a da soma dos \u00e2ngulos internos de um tri\u00e2ngulo para encontrar \u03b1.<\/p><p> <\/p><figure><img decoding=\"async\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2023\/10\/image-17.png\" alt=\"\" \/><\/figure><p> <\/p><p>\u03b1 = 180\u00b0 &#8211; arccos(\u221a6\/3) &#8211; arccos(\u221a6\/3) ~ 109.47\u00b0<\/p><p> <\/p><p>\u00a0<\/p><p> <\/p><p>Ufa! Acho que isso explica porque geralmente n\u00e3o vemos essa explica\u00e7\u00e3o \ud83d\ude42<\/p><p><\/p><div>\u00a0<\/div><div>\u00a0<\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-9775d80 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"9775d80\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2c2e5ad jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"2c2e5ad\" 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<h1>Sobre o autor<\/h1><p class=\"eplus-wrapper\">Podem me chamar de\u00a0<strong>Zero<\/strong>, fiz Licenciatura em Matem\u00e1tica pela USP, Mestrado na UNESP sobre a aprendizagem de Pensamento Computacional, Doutorado na UNICAMP sobre a aprendizagem de Demonstra\u00e7\u00f5es Matem\u00e1ticas, Especializa\u00e7\u00e3o em Inform\u00e1tica Aplicada \u00e0 Educa\u00e7\u00e3o no IFRJ e atualmente sou docente de Matem\u00e1tica e curso Licenciatura em Qu\u00edmica, ambos no IFRJ.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-23788f7f e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"23788f7f\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-7dda89a1 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"7dda89a1\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-d789714 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"d789714\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-1f2219b jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"1f2219b\" 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>Como citar:&nbsp;&nbsp;<\/p>\n<p>&nbsp;SILVA, Marcos Henrique de Paula Dias da. (2023). Porque o \u00e2ngulo da liga\u00e7\u00e3o tetra\u00e9drica \u00e9 109\u00b0? <strong>Revista Blogs Unicamp<\/strong>, V.9, N.2. Dispon\u00edvel em: <a href=\"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/porque-o-angulo-da-ligacao-tetraedrica-e-109\/\">https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/porque-o-angulo-da-ligacao-tetraedrica-e-109\/<\/a><a href=\"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/porque-o-angulo-da-ligacao-tetraedrica-e-109\/\"><br><\/a>Acesso em dd\/mm\/aaaa.<\/p>\n<p><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-25f4bb0d e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"25f4bb0d\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-f3af357 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"f3af357\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-4241ebf6 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"4241ebf6\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-db1d0fc jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"db1d0fc\" 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><strong>Sobre a imagem destacada:<\/strong><\/p>\n<p><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Arte Tess AI&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:897,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:1,&quot;11&quot;:4,&quot;12&quot;:0}\">Imagem dispon\u00edvel no Freepik, editada por <a href=\"https:\/\/clorofreela.com\" target=\"_blank\" rel=\"noopener\">Clorofreela<\/a>.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-1c7c7159 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"1c7c7159\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t<div class=\"elementor-element elementor-element-32ade250 e-con-full e-flex jltma-glass-effect-no e-con e-child\" data-id=\"32ade250\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-3e493613 jltma-glass-effect-no elementor-widget elementor-widget-ha-link-hover happy-addon ha-link-hover\" data-id=\"3e493613\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"ha-link-hover.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t        <link-hover class=\"ha_content__item\">\n            <a href=\"https:\/\/www.blogs.unicamp.br\/revista\/edicao\/v-9-n-2-2023\/\" class=\"ha-link ha-link--kale\" rel=\"nofollow\">Mais postagens &lt;br&gt;desta edi\u00e7\u00e3o<\/a>\n        <\/link-hover>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7363d041 premium-blog-align-left jltma-glass-effect-no elementor-widget elementor-widget-premium-addon-blog\" data-id=\"7363d041\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_blog_columns_number&quot;:&quot;100%&quot;,&quot;premium_blog_grid&quot;:&quot;yes&quot;,&quot;premium_blog_layout&quot;:&quot;even&quot;,&quot;premium_blog_columns_number_tablet&quot;:&quot;50%&quot;,&quot;premium_blog_columns_number_mobile&quot;:&quot;100%&quot;,&quot;scroll_to_offset&quot;:&quot;yes&quot;,&quot;filter_flag&quot;:&quot;cat&quot;}\" data-widget_type=\"premium-addon-blog.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t\t\t\t\t\t<div class=\"premium-blog-filter\">\n\t\t\t<ul class=\"premium-blog-filters-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<li>\n\t\t\t\t\t\t\t<a href=\"javascript:;\" class=\"category active\" data-filter=\"v9-n2-2013\">\n\t\t\t\t\t\t\t\tV.9 N. 2 2023\t\t\t\t\t\t\t<\/a>\n\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t<\/ul>\n\t\t<\/div>\n\t\t\t\t\n\n\t\t\t\t<div class=\"premium-blog-wrap  premium-blog-even\" data-page=\"12180\">\n\t\t\t\t\t\t\t\t\t<div class=\"premium-blog-post-outer-container artigo engenharias v9-n2-2013\" data-total=\"3\">\n\t\t\t<div class=\"premium-blog-post-container premium-blog-skin-classic\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-blog-content-wrapper empty-thumb\">\n\n\t\t\t\t\t<div class=\"premium-blog-inner-container\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h2 class=\"premium-blog-entry-title\">\n\t\t\t<a href=\"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/os-povos-indigenas-ameacados-por-uma-empresa-auxiliar-dos-carros-eletricos\/\" target=\"_self\">\n\t\t\t\tOs povos ind\u00edgenas amea\u00e7ados por uma empresa auxiliar dos carros el\u00e9tricos?\t\t\t<\/a>\n\t\t<\/h2>\n\t\t\t\t<div class=\"premium-blog-entry-meta\">\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\t\t\t\t<div class=\"premium-blog-post-outer-container artigo ciencias-biologicas v9-n2-2013\" data-total=\"3\">\n\t\t\t<div class=\"premium-blog-post-container premium-blog-skin-classic\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-blog-content-wrapper empty-thumb\">\n\n\t\t\t\t\t<div class=\"premium-blog-inner-container\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h2 class=\"premium-blog-entry-title\">\n\t\t\t<a href=\"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/cobertura-sobre-mudancas-climaticas-e-distante-tecnica-e-limitada-diz-estudo\/\" target=\"_self\">\n\t\t\t\tCobertura sobre mudan\u00e7as clim\u00e1ticas \u00e9 distante, t\u00e9cnica e limitada, diz estudo\t\t\t<\/a>\n\t\t<\/h2>\n\t\t\t\t<div class=\"premium-blog-entry-meta\">\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\t\t\t\t<div class=\"premium-blog-post-outer-container artigo ciencias-exatas-e-da-terra-2 v9-n2-2013 zero\" data-total=\"3\">\n\t\t\t<div class=\"premium-blog-post-container premium-blog-skin-classic\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-blog-content-wrapper empty-thumb\">\n\n\t\t\t\t\t<div class=\"premium-blog-inner-container\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h2 class=\"premium-blog-entry-title\">\n\t\t\t<a href=\"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/o-problema-agua-luz-e-esgoto\/\" target=\"_self\">\n\t\t\t\tO problema \u00c1gua, Luz e Esgoto\t\t\t<\/a>\n\t\t<\/h2>\n\t\t\t\t<div class=\"premium-blog-entry-meta\">\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\t\t\t\t<div class=\"premium-blog-post-outer-container artigo ciencias-humandas pemcie v9-n2-2013\" data-total=\"3\">\n\t\t\t<div class=\"premium-blog-post-container premium-blog-skin-classic\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-blog-content-wrapper empty-thumb\">\n\n\t\t\t\t\t<div class=\"premium-blog-inner-container\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h2 class=\"premium-blog-entry-title\">\n\t\t\t<a href=\"https:\/\/www.blogs.unicamp.br\/revista\/2023\/11\/29\/a-permissividade-dos-discursos-do-trote-caso-unisa\/\" target=\"_self\">\n\t\t\t\tA permissividade dos discursos do trote: caso UNISA\t\t\t<\/a>\n\t\t<\/h2>\n\t\t\t\t<div class=\"premium-blog-entry-meta\">\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\t<\/div>\n\t\t\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\n\t\t\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Como podemos obter o \u00e2ngulo da liga\u00e7\u00e3o tetra\u00e9drica? Conhe\u00e7a o processo matem\u00e1tico por tr\u00e1s desse intrigante mist\u00e9rio da Qu\u00edmica!<\/p>\n","protected":false},"author":362,"featured_media":12365,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_header_footer","format":"standard","meta":{"_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":[1165,891,1166,894],"tags":[1216,102,1215,895],"class_list":["post-12180","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-artigo","category-ciencias-exatas-e-da-terra","category-v9-n2-2013","category-zero","tag-angulo","tag-matematica","tag-tetraedro","tag-zero"],"jetpack_featured_media_url":"https:\/\/www.blogs.unicamp.br\/revista\/wp-content\/uploads\/sites\/279\/2023\/11\/Porque_o_a_ngulo_da_ligac_a_o_tetrae_drica_e_109_1.webp","_links":{"self":[{"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/posts\/12180","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/users\/362"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/comments?post=12180"}],"version-history":[{"count":22,"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/posts\/12180\/revisions"}],"predecessor-version":[{"id":12458,"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/posts\/12180\/revisions\/12458"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/media\/12365"}],"wp:attachment":[{"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/media?parent=12180"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/categories?post=12180"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/revista\/wp-json\/wp\/v2\/tags?post=12180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}