{"id":7392,"date":"2012-02-12T21:45:35","date_gmt":"2012-02-12T21:45:35","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7392"},"modified":"2022-01-16T21:59:07","modified_gmt":"2022-01-16T21:59:07","slug":"um-hexagono-regular-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7392","title":{"rendered":"Um hex\u00e1gono regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_7392' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7392' class='GTTabs_curr'><a  id=\"7392_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7392' ><a  id=\"7392_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_7392'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag37-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7393\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7393\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag37-2.jpg\" data-orig-size=\"244,249\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Hex\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag37-2.jpg\" class=\"alignright  wp-image-7393\" title=\"Hex\u00e1gono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag37-2.jpg\" alt=\"\" width=\"146\" height=\"149\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag37-2.jpg 244w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag37-2-146x150.jpg 146w\" sizes=\"auto, (max-width: 146px) 100vw, 146px\" \/><\/a>A figura ao lado \u00e9 um hex\u00e1gono regular.<\/p>\n<p>Calcula a sua \u00e1rea, sabendo que o raio da circunfer\u00eancia \u00e9 4 cm.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7392' onClick='GTTabs_show(1,7392)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7392'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><span style=\"text-decoration: underline;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7394\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7394\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\" data-orig-size=\"262,274\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Hexagono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\" class=\"alignright size-full wp-image-7394\" title=\"Hexagono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\" alt=\"\" width=\"262\" height=\"274\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png 262w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2-143x150.png 143w\" sizes=\"auto, (max-width: 262px) 100vw, 262px\" \/><\/a>Comecemos por provar que o tri\u00e2ngulo [AOB] \u00e9 equil\u00e1tero<\/span>:<\/p>\n<p>[OA] e [OB] s\u00e3o raios da mesma circunfer\u00eancia, logo o tri\u00e2ngulo [AOB] \u00e9 is\u00f3sceles.<\/p>\n<p>Num tri\u00e2ngulo, a lados geometricamente iguais op\u00f5em-se \u00e2ngulos geometricamente iguais. Logo, os \u00e2ngulos OAB e OBA s\u00e3o geometricamente iguais.<\/p>\n<p>Como a amplitude do \u00e2ngulo ao centro AOB \u00e9 60\u00ba, pois \u00e9 igual \u00e0 amplitude de cada um dos 6 arcos geometricamente iguais em que a circunfer\u00eancia foi dividida, resulta que os tr\u00eas \u00e2ngulos internos do tri\u00e2ngulo [AOB] s\u00e3o geometricamente iguais.<\/p>\n<p>Sendo equi\u00e2ngulo, o tri\u00e2ngulo [AOB] \u00e9 equil\u00e1tero.<\/p>\n<\/p>\n<p><span style=\"text-decoration: underline;\">Calculemos agora o ap\u00f3tema<\/span>:<\/p>\n<p>Aplicando o teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [AOM], temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\overline {OM} }&amp; = &amp;{\\sqrt {{{\\overline {AO} }^2} &#8211; {{\\overline {AM} }^2}} } \\\\<br \/>\n{}&amp; = &amp;{\\sqrt {{4^2} &#8211; {2^2}} } \\\\<br \/>\n{}&amp; = &amp;{\\sqrt {12} } \\\\<br \/>\n{}&amp; = &amp;{2\\sqrt 3 }<br \/>\n\\end{array}$$<\/p>\n<p>Assim, a \u00e1rea do tri\u00e2ngulo [AOB] \u00e9:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{A_{[AOB]}}}&amp; = &amp;{\\frac{{\\overline {AB}\u00a0 \\times \\overline {OM} }}{2}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{4 \\times 2\\sqrt 3 }}{2}} \\\\<br \/>\n{}&amp; = &amp;{4\\sqrt 3 \\,\\,c{m^2}}<br \/>\n\\end{array}$$<\/p>\n<p>Logo, a \u00e1rea do hex\u00e1gono \u00e9:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\nA&amp; = &amp;{6 \\times {A_{[AOB]}}} \\\\<br \/>\n{}&amp; = &amp;{6 \\times 4\\sqrt 3 } \\\\<br \/>\n{}&amp; = &amp;{24\\sqrt 3 \\,\\,c{m^2}}<br \/>\n\\end{array}$$<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7392' onClick='GTTabs_show(0,7392)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A figura ao lado \u00e9 um hex\u00e1gono regular. Calcula a sua \u00e1rea, sabendo que o raio da circunfer\u00eancia \u00e9 4 cm. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":20423,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,278],"tags":[426,108,283],"series":[],"class_list":["post-7392","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-circunferencia-e-poligonos","tag-9-o-ano","tag-area","tag-poligono-regular"],"views":4591,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/9V1Pag037-2_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7392","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=7392"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7392\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20423"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7392"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}