{"id":3756,"date":"2010-10-08T03:03:20","date_gmt":"2010-10-08T02:03:20","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=3756"},"modified":"2022-01-13T14:32:20","modified_gmt":"2022-01-13T14:32:20","slug":"rascunho-4","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=3756","title":{"rendered":"Um pol\u00edgono regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_3756' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_3756' class='GTTabs_curr'><a  id=\"3756_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_3756' ><a  id=\"3756_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_3756'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Qual a medida, em fun\u00e7\u00e3o da metade do \u00e2ngulo ao centro que lhe corresponde, o lado de um tri\u00e2ngulo equil\u00e1tero inscrito numa circunfer\u00eancia de raio <em><strong>r<\/strong><\/em>? E do quadrado? E do pent\u00e1gono regular? E do pol\u00edgono regular de <em><strong>n<\/strong><\/em> lados?<\/li>\n<li>Determine, em fun\u00e7\u00e3o da metade do \u00e2ngulo ao centro correspondente ao lado, o ap\u00f3tema e a \u00e1rea de um pol\u00edgono regular de <em><strong>n<\/strong><\/em> lados inscrito numa circunfer\u00eancia de raio <em><strong>r<\/strong><\/em>.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_3756' onClick='GTTabs_show(1,3756)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_3756'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><strong>1.\u00a0<\/strong><\/p>\n<p style=\"padding-left: 30px;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script>Seja $\\beta =2\\alpha $ a amplitude do \u00e2ngulo ao centro correspondente a um lado de um pol\u00edgono regular de $n$ lados, inscrito numa circunfer\u00eancia de raio $r$. Assim, ser\u00e1: \\[\\beta =2\\alpha =\\frac{360{}^\\text{o}}{n}\\]<\/p>\n<p style=\"padding-left: 30px;\"><strong>Tri\u00e2ngulo equil\u00e1tero<\/strong>:<\/p>\n<p style=\"padding-left: 30px;\">No caso do tri\u00e2ngulo equil\u00e1tero, ser\u00e1 $\\alpha =\\frac{360{}^\\text{o}}{2\\times 3}=60{}^\\text{o}$. Como\u00a0\\[sen\\,60{}^\\text{o}=\\frac{\\frac{l}{2}}{r}\\Leftrightarrow \\frac{l}{2}=r\\times sen\\,60{}^\\text{o}\\] ent\u00e3o \\[{{l}_{T}}=2r\\times sen\\,\\frac{120{}^\\text{o}}{2}\\]<\/p>\n<p style=\"padding-left: 30px;\"><strong>Quadrado<\/strong>:<\/p>\n<p style=\"padding-left: 30px;\">No caso do quadrado, ser\u00e1 $\\alpha =\\frac{360{}^\\text{o}}{2\\times 4}=45{}^\\text{o}$. Como\u00a0\\[sen\\,45{}^\\text{o}=\\frac{\\frac{l}{2}}{r}\\Leftrightarrow \\frac{l}{2}=r\\times sen\\,45{}^\\text{o}\\] ent\u00e3o \\[{{l}_{Q}}=2r\\times sen\\,\\frac{90{}^\\text{o}}{2}\\]<\/p>\n<p style=\"padding-left: 30px;\"><strong>Pent\u00e1gono regular<\/strong>:<\/p>\n<p style=\"padding-left: 30px;\">No caso do pent\u00e1gono regular, ser\u00e1 $\\alpha =\\frac{360{}^\\text{o}}{2\\times 5}=36{}^\\text{o}$. Como\u00a0\\[sen\\,36{}^\\text{o}=\\frac{\\frac{l}{2}}{r}\\Leftrightarrow \\frac{l}{2}=r\\times sen\\,36{}^\\text{o}\\] ent\u00e3o \\[{{l}_{P}}=2r\\times sen\\,\\frac{72{}^\\text{o}}{2}\\]<\/p>\n<p style=\"padding-left: 30px;\"><strong>Pol\u00edgono regular de <em>n<\/em> lados<\/strong>:<\/p>\n<p style=\"padding-left: 30px;\">No caso do pol\u00edgono regular de <em>n<\/em> lados, ser\u00e1 $\\alpha =\\frac{360{}^\\text{o}}{2\\times n}$. Como\u00a0\\[sen\\,\\frac{360{}^\\text{o}}{2n}=\\frac{\\frac{l}{2}}{r}\\Leftrightarrow \\frac{l}{2}=r\\times sen\\,\\frac{360{}^\\text{o}}{2n}\\] ent\u00e3o \\[{{l}_{n}}=2r\\times sen\\,\\frac{360{}^\\text{o}}{2n}\\]<\/p>\n<p><strong>2.<\/strong><\/p>\n<p style=\"padding-left: 30px;\"><strong>Ap\u00f3tema<\/strong>:<\/p>\n<p style=\"padding-left: 30px;\">\u00a0Considerando o tri\u00e2ngulo ret\u00e2ngulo correspondente, temos \\[\\cos \\frac{360{}^\\text{o}}{2n}=\\frac{ap}{r}\\] donde \\[ap=r\\times \\cos \\,\\frac{360{}^\\text{o}}{2n}\\]<\/p>\n<p style=\"padding-left: 30px;\"><strong>\u00c1rea<\/strong>:<\/p>\n<p style=\"padding-left: 30px;\">\u00a0A \u00e1rea do\u00a0tri\u00e2ngulo [AOB], isto \u00e9, a \u00e1rea de\u00a0$\\frac{1}{n}$\u00a0da \u00e1rea desse pol\u00edgono regular de <em>n<\/em> lados, \u00e9 dada por: \\[{{A}_{[AOB]}}=\\frac{l\\times ap}{2}=\\frac{2r\\times sen\\,\\frac{360{}^\\text{o}}{2n}\\times r\\times \\cos \\,\\frac{360{}^\\text{o}}{2n}}{2}={{r}^{2}}\\times sen\\,\\frac{360{}^\\text{o}}{2n}\\times \\cos \\,\\frac{360{}^\\text{o}}{2n}\\]<\/p>\n<p style=\"padding-left: 30px;\">Logo, a \u00e1rea desse pol\u00edgono regular de <em>n<\/em> lados \u00e9 dada por: \\[{{A}_{[Pn]}}=n{{r}^{2}}\\times sen\\,\\frac{360{}^\\text{o}}{2n}\\times \\cos \\,\\frac{360{}^\\text{o}}{2n}\\]<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_3756' onClick='GTTabs_show(0,3756)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Qual a medida, em fun\u00e7\u00e3o da metade do \u00e2ngulo ao centro que lhe corresponde, o lado de um tri\u00e2ngulo equil\u00e1tero inscrito numa circunfer\u00eancia de raio r? E do quadrado? E do&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19459,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,99],"tags":[422,423],"series":[],"class_list":["post-3756","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-trigonometria","tag-11-o-ano","tag-trigonometria"],"views":2439,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/Poligonos_regulares.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3756","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3756"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3756\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19459"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3756"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3756"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3756"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=3756"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}