{"id":13767,"date":"2018-03-04T16:38:49","date_gmt":"2018-03-04T16:38:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13767"},"modified":"2022-01-06T14:53:19","modified_gmt":"2022-01-06T14:53:19","slug":"uma-piramide-triangular-regular-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13767","title":{"rendered":"Uma pir\u00e2mide triangular regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_13767' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13767' class='GTTabs_curr'><a  id=\"13767_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13767' ><a  id=\"13767_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_13767'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/PiramideTriangular2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13769\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13769\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/PiramideTriangular2.png\" data-orig-size=\"275,245\" 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=\"Piramide Triangular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/PiramideTriangular2.png\" class=\"alignright wp-image-13769\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/PiramideTriangular2.png\" alt=\"\" width=\"240\" height=\"214\" \/><\/a>O per\u00edmetro da base de uma pir\u00e2mide triangular regular (pir\u00e2mide cuja base \u00e9 um tri\u00e2ngulo equil\u00e1tero) \u00e9 igual a 24 cm. A altura da face lateral da pir\u00e2mide \u00e9 igual ao dobro da aresta da base <del>e a altura da base mede aproximadamente 6,9 cm<\/del>.<\/p>\n<p>Determina:<\/p>\n<ol>\n<li>a \u00e1rea da superf\u00edcie de uma das faces laterais;<\/li>\n<li>a \u00e1rea da base;<\/li>\n<li>a \u00e1rea da superf\u00edcie da pir\u00e2mide.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13767' onClick='GTTabs_show(1,13767)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13767'>\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\/2017-18-MF9P2-pag25-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13772\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13772\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-3.png\" data-orig-size=\"1272,1125\" 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=\"Planifica\u00e7\u00e3o da pir\u00e2mide\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-3-1024x906.png\" class=\"alignright wp-image-13772\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-3-300x265.png\" alt=\"\" width=\"360\" height=\"318\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-3-300x265.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-3-768x679.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-3-1024x906.png 1024w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-3.png 1272w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/><\/a>Comecemos por determinar o valor exato da altura da base da pir\u00e2mide.\u00a0Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [AEC], vem:<\/p>\n<p>\\[\\overline {CE} = \\sqrt {{{\\overline {AC} }^2} &#8211; {{\\overline {AE} }^2}} = \\sqrt {{8^2} &#8211; {4^2}} = \\sqrt {48} = 4\\sqrt 3 \\]<\/p>\n<p>Portanto,\u00a0\\(\\overline {CE} = 4\\sqrt 3 \\approx 6,9\\) cm, conforme indicado no enunciado.<\/p>\n<ol>\n<li>A \u00e1rea\u00a0da superf\u00edcie de uma das faces laterais \u00e9 \\({A_{fL}} = \\frac{{\\overline {BC} \\times \\overline {DV} }}{2} = \\frac{{8 \\times 16}}{2} = 64\\) cm<sup>2<\/sup>.<br \/>\n\u00ad<\/li>\n<li>A \u00e1rea da base \u00e9\u00a0\\({A_B} = \\frac{{\\overline {AB} \\times \\overline {CE} }}{2} = \\frac{{8 \\times 4\\sqrt 3 }}{2} = 16\\sqrt 3 \\approx 27,7\\) cm<sup>2<\/sup>.<br \/>\n\u00ad<\/li>\n<li>A\u00a0\u00e1rea da superf\u00edcie da pir\u00e2mide \u00e9\u00a0\\({A_T} = {A_B} + 3 \\times {A_{fL}} = 16\\sqrt 3 + 3 \\times 64 = 192 + 16\\sqrt 3 \\approx 219,7\\)\u00a0 cm<sup>2<\/sup>.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13767' onClick='GTTabs_show(0,13767)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O per\u00edmetro da base de uma pir\u00e2mide triangular regular (pir\u00e2mide cuja base \u00e9 um tri\u00e2ngulo equil\u00e1tero) \u00e9 igual a 24 cm. A altura da face lateral da pir\u00e2mide \u00e9 igual ao&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":13768,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,108,471],"series":[],"class_list":["post-13767","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-piramide"],"views":3527,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/PiramideTriangular.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13767","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=13767"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13767\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/13768"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13767"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13767"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13767"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13767"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}