{"id":13510,"date":"2018-02-08T17:22:40","date_gmt":"2018-02-08T17:22:40","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13510"},"modified":"2022-01-16T15:54:01","modified_gmt":"2022-01-16T15:54:01","slug":"uma-piramide-quadrangular","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13510","title":{"rendered":"Uma pir\u00e2mide quadrangular"},"content":{"rendered":"<p><ul id='GTTabs_ul_13510' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13510' class='GTTabs_curr'><a  id=\"13510_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13510' ><a  id=\"13510_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_13510'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13511\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13511\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1.png\" data-orig-size=\"265,275\" 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=\"Pir\u00e2mide quadrangular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1.png\" class=\"alignright size-full wp-image-13511\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1.png\" alt=\"\" width=\"265\" height=\"275\" \/><\/a>Considera a pir\u00e2mide quadrangular [ABCDE] representada na figura.<\/p>\n<p>Sabe-se que [DB] \u00e9 a diagonal do quadril\u00e1tero [ABCD] e que F \u00e9 a proje\u00e7\u00e3o ortogonal de E no plano que cont\u00e9m a base da pir\u00e2mide.<\/p>\n<p>Utilizando uma decomposi\u00e7\u00e3o em pir\u00e2mides triangulares, verifica que o volume da pir\u00e2mide quadrangular \u00e9 igual a um ter\u00e7o do produto da \u00e1rea da base pela altura.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13510' onClick='GTTabs_show(1,13510)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13510'>\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\/02\/9V2Pag019-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13511\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13511\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1.png\" data-orig-size=\"265,275\" 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=\"Pir\u00e2mide quadrangular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1.png\" class=\"alignright size-full wp-image-13511\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1.png\" alt=\"\" width=\"265\" height=\"275\" \/><\/a>Consideremos a pir\u00e2mide [ABCDE], de volume V, decomposta nas pir\u00e2mides [ABDE] e [BCDE], de volumes V<sub>1<\/sub> e V<sub>2<\/sub>, respetivamente.<\/p>\n<p>Tem-se, portanto,\u00a0\\(V = {V_1} + {V_2}\\), donde se obt\u00e9m sucessivamente:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}V&amp; = &amp;{{V_1} + {V_2}}\\\\{}&amp; = &amp;{\\frac{{{A_{\\left[ {ABD} \\right]}} \\times \\overline {EF} }}{3} + \\frac{{{A_{\\left[ {BCD} \\right]}} \\times \\overline {EF} }}{3}}\\\\{}&amp; = &amp;{\\left( {\\frac{{{A_{\\left[ {ABD} \\right]}}}}{3} + \\frac{{{A_{\\left[ {BCD} \\right]}}}}{3}} \\right) \\times \\overline {EF} }\\\\{}&amp; = &amp;{\\frac{{{A_{\\left[ {ABD} \\right]}} + {A_{\\left[ {BCD} \\right]}}}}{3} \\times \\overline {EF} }\\\\{}&amp; = &amp;{\\frac{{{A_{\\left[ {ABCD} \\right]}} \\times \\overline {EF} }}{3}}\\end{array}\\]<\/p>\n<p>Portanto,\u00a0o volume da pir\u00e2mide quadrangular \u00e9 igual a um ter\u00e7o do produto da \u00e1rea da base pela altura:\u00a0\\[V = \\frac{{{A_{\\left[ {ABCD} \\right]}} \\times \\overline {EF} }}{3}\\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13510' onClick='GTTabs_show(0,13510)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considera a pir\u00e2mide quadrangular [ABCDE] representada na figura. Sabe-se que [DB] \u00e9 a diagonal do quadril\u00e1tero [ABCD] e que F \u00e9 a proje\u00e7\u00e3o ortogonal de E no plano que cont\u00e9m a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20374,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,471,109],"series":[],"class_list":["post-13510","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-piramide","tag-volume"],"views":4138,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V2Pag019-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13510","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=13510"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13510\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20374"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13510"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13510"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13510"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13510"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}