{"id":13778,"date":"2018-03-04T22:30:46","date_gmt":"2018-03-04T22:30:46","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13778"},"modified":"2022-01-16T18:23:58","modified_gmt":"2022-01-16T18:23:58","slug":"uma-piramide-quadrangular-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13778","title":{"rendered":"Uma pir\u00e2mide quadrangular"},"content":{"rendered":"<p><ul id='GTTabs_ul_13778' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13778' class='GTTabs_curr'><a  id=\"13778_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13778' ><a  id=\"13778_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_13778'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Pir\u00e2mideQuadrangular2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13780\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13780\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Pir\u00e2mideQuadrangular2.png\" data-orig-size=\"350,285\" 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\/03\/Pir\u00e2mideQuadrangular2.png\" class=\"alignright size-medium wp-image-13780\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Pir\u00e2mideQuadrangular2-300x244.png\" alt=\"\" width=\"300\" height=\"244\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Pir\u00e2mideQuadrangular2-300x244.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Pir\u00e2mideQuadrangular2.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>A \u00e1rea da base de uma pir\u00e2mide quadrangular regular \u00e9 igual a 25 cm<sup>2<\/sup>.<\/p>\n<p>A altura de cada face lateral \u00e9 4 cm <del>e a altura da pir\u00e2mide \u00e9, aproximadamente, 3,1 cm<\/del>.<\/p>\n<p>Determina:<\/p>\n<ol>\n<li>o volume da pir\u00e2mide;<\/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_13778' onClick='GTTabs_show(1,13778)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13778'>\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-pag25-4.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13784\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13784\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-4.png\" data-orig-size=\"505,532\" 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-pag25-4.png\" class=\"alignright wp-image-13784\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-4-285x300.png\" alt=\"\" width=\"400\" height=\"421\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-4-285x300.png 285w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag25-4.png 505w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a>Comecemos por determinar a altura da pir\u00e2mide. No espa\u00e7o e por aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [OEV[, temos:<\/p>\n<p>\\[\\overline {OV} = \\sqrt {{{\\overline {EV} }^2} &#8211; {{\\overline {OE} }^2}} = \\sqrt {{4^2} &#8211; {{\\left( {\\frac{5}{2}} \\right)}^2}} = \\sqrt {\\frac{{39}}{4}} = \\frac{{\\sqrt {39} }}{2}\\]<\/p>\n<p>Portanto, \\(\\overline {OV} = \\frac{{\\sqrt {39} }}{2} \\approx 3,1\\) cm, como indicado no enunciado.<\/p>\n<ol>\n<li>A pir\u00e2mide tem, aproximadamente, 26,0 cm<sup>3<\/sup> de volume:<br \/>\n\\[\\begin{array}{*{20}{l}}V&amp; = &amp;{\\frac{1}{3} \\times \\left( {\\overline {AB} \\times \\overline {BC} } \\right) \\times \\overline {OV} }\\\\{}&amp; = &amp;{\\frac{1}{3} \\times \\left( {5 \\times 5} \\right) \\times \\frac{{\\sqrt {39} }}{2}}\\\\{}&amp; = &amp;{\\frac{{25\\sqrt {39} }}{6}}\\\\{}&amp; \\approx &amp;{26,0}\\end{array}\\]<\/li>\n<li>A \u00e1rea da superf\u00edcie da pir\u00e2mide \u00e9 65 cm<sup>2<\/sup>:<br \/>\n\\[{A_T} = \\left( {\\overline {AB} \\times \\overline {BC} } \\right) + 4 \\times \\left( {\\frac{{\\overline {CD} \\times \\overline {VE} }}{2}} \\right) = \\left( {5 \\times 5} \\right) + 4 \\times \\left( {\\frac{{5 \\times 4}}{2}} \\right) = 65\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13778' onClick='GTTabs_show(0,13778)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A \u00e1rea da base de uma pir\u00e2mide quadrangular regular \u00e9 igual a 25 cm2. A altura de cada face lateral \u00e9 4 cm e a altura da pir\u00e2mide \u00e9, aproximadamente, 3,1&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20396,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,108,483,109],"series":[],"class_list":["post-13778","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-pramide","tag-volume"],"views":2983,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag025-4_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13778","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=13778"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13778\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20396"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13778"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13778"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13778"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}