{"id":13882,"date":"2018-03-07T18:18:28","date_gmt":"2018-03-07T18:18:28","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13882"},"modified":"2022-01-06T18:10:39","modified_gmt":"2022-01-06T18:10:39","slug":"um-prisma-hexagonal-regular","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13882","title":{"rendered":"Um prisma hexagonal regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_13882' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13882' class='GTTabs_curr'><a  id=\"13882_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13882' ><a  id=\"13882_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_13882'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13883\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13883\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\" data-orig-size=\"165,245\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Prisma hexagonal\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\" class=\"alignright size-full wp-image-13883\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\" alt=\"\" width=\"165\" height=\"245\" \/><\/a>A base de um prisma hexagonal regular est\u00e1 inscrita num c\u00edrculo de 8 cm de raio.<br \/>\nA altura do prisma \u00e9 igual ao di\u00e2metro do c\u00edrculo.<\/p>\n<p>Determina:<\/p>\n<ol>\n<li>a \u00e1rea das seis faces laterais do prisma;<\/li>\n<li>a \u00e1rea da superf\u00edcie do prisma;<\/li>\n<li>o volume do prisma.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13882' onClick='GTTabs_show(1,13882)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13882'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13883\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13883\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\" data-orig-size=\"165,245\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Prisma hexagonal\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\" class=\"alignright size-full wp-image-13883\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12b.png\" alt=\"\" width=\"165\" height=\"245\" \/><\/a>Seja <em>O<\/em> o centro da base do prisma que, como se sabe, pode ser decomposta em seis tri\u00e2ngulos equil\u00e1teros geometricamente iguais.<\/p>\n<p>Seja <em>M<\/em> o ponto m\u00e9dio do segmento de reta [<em>AB<\/em>].<\/p>\n<p>Determinemos o comprimento da altura [<em>OM<\/em>] do tri\u00e2ngulo equil\u00e1tero [<em>OAB<\/em>]:<\/p>\n<p>\\[\\overline {OM} = \\sqrt {{{\\overline {AO} }^2} &#8211; {{\\overline {AM} }^2}} = \\sqrt {{8^2} &#8211; {4^2}} = \\sqrt {48} = 4\\sqrt 3 \\]<br \/>\n\u00ad<\/p>\n<ol>\n<li>A\u00a0\u00e1rea das seis faces laterais do prisma \u00e9\u00a0\\({A_L} = 6 \\times \\left( {\\overline {AB} \\times \\overline {AP} } \\right) = 6 \\times \\left( {8 \\times 16} \\right) = 768\\)\u00a0 cm<sup>2<\/sup>.<br \/>\n\u00ad<\/li>\n<li>A\u00a0\u00e1rea da superf\u00edcie do prisma \u00e9 \\({A_T} = 2 \\times {A_B} + {A_L} = 2 \\times \\left( {6 \\times \\frac{{\\overline {AB} \\times \\overline {OM} }}{2}} \\right) + 768 = 2 \\times \\left( {6 \\times \\frac{{8 \\times 4\\sqrt 3 }}{2}} \\right) + 768 = \\left( {768 + 192\\sqrt 3 } \\right)\\) \u00a0cm<sup>2<\/sup>.<br \/>\n\u00ad<\/li>\n<li>O prisma tem\u00a0\\(V = 6 \\times \\frac{{\\overline {AB} \\times \\overline {OM} }}{2} \\times \\overline {AP} = 6 \\times \\frac{{8 \\times 4\\sqrt 3 }}{2} \\times 16 = 1536\\sqrt 3 \\)\u00a0 cm<sup>3<\/sup> de volume.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13882' onClick='GTTabs_show(0,13882)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A base de um prisma hexagonal regular est\u00e1 inscrita num c\u00edrculo de 8 cm de raio. A altura do prisma \u00e9 igual ao di\u00e2metro do c\u00edrculo. Determina: a \u00e1rea das seis&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":13884,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,108,468,109],"series":[],"class_list":["post-13882","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-distancias-areas-e-volumes-de-solidos","tag-9-o-ano","tag-area","tag-prisma","tag-volume"],"views":7965,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag032-12.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13882","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=13882"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13882\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/13884"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13882"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13882"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13882"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}