{"id":4961,"date":"2016-10-19T17:46:26","date_gmt":"2016-10-19T15:46:26","guid":{"rendered":"http:\/\/pimedios.es\/?p=4961"},"modified":"2016-10-19T17:46:26","modified_gmt":"2016-10-19T15:46:26","slug":"el-volumen-del-tetraedro","status":"publish","type":"post","link":"https:\/\/pimedios.jesussoto.es\/?p=4961","title":{"rendered":"El volumen del tetraedro"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/9e\/120px-Tetrahedron-slowturn.gif\" alt=\"\" width=\"120\" height=\"120\" \/>Un tetraedro es una pir\u00e1mide de base triangular. Como figura geom\u00e9trica hablamos de un poliedro de cuatro caras triangulares. El tetraedro es un de los s\u00f3lidos plat\u00f3nicos, cuando los tri\u00e1ngulos son equil\u00e1teros, siendo conocido desde la Grecia cl\u00e1sica. As\u00ed que el volumen era conocido, siendo $$V=\\frac{1}{3}hA,$$ donde $h$ es la altura y $A$ el \u00e1rea de la base.<\/p>\n<figure style=\"width: 223px\" class=\"wp-caption alignleft\"><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Triangle_with_notations_2.svg#\/media\/File:Triangle_with_notations_2.svg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/4\/49\/Triangle_with_notations_2.svg\/1200px-Triangle_with_notations_2.svg.png\" alt=\"Triangle with notations 2.svg\" width=\"223\" height=\"134\" \/><\/a><figcaption class=\"wp-caption-text\"><br \/><span style=\"font-size: smaller;\">De David Weisman (<a class=\"extiw\" title=\"en:User:Dweisman\" href=\"https:\/\/en.wikipedia.org\/wiki\/User:Dweisman\">Dweisman<\/a>) &#8211; En-Wiki. Original description is\/was <a class=\"extiw\" title=\"en:Image:TriangleWithNotations.svg\" href=\"https:\/\/en.wikipedia.org\/wiki\/Image:TriangleWithNotations.svg\">here<\/a>, Dominio p\u00fablico, <a href=\"https:\/\/commons.wikimedia.org\/w\/index.php?curid=2596911\">https:\/\/commons.wikimedia.org\/w\/index.php?curid=2596911<\/a><\/span><\/figcaption><\/figure>\n<p>Her\u00f3n de Alejandr\u00eda(siglo I d. C.) hab\u00eda encontrado una f\u00f3rmula para determinar el \u00e1rea de un tri\u00e1ngulo:<br \/>\n$$A=\\sqrt{s(s-a)(s-b)(s-c)}$$<\/p>\n<p>donde $s=\\frac{a+b+c}{2}$, el semiper\u00edmetro del tri\u00e1ngulo. Esta f\u00f3rmula la prob\u00f3 en su libro <em>Metrica<\/em>, escrito sobre el 60 de nuestra era. Es probable que Arqu\u00edmedes la conociera y Heron simplemente la recogi\u00f3.<\/p>\n<p>Siglos despu\u00e9s Tartaglia, a mediados del siglo XVI, extender\u00eda la f\u00f3rmula de Her\u00f3n (ver <a href=\"https:\/\/en.wikipedia.org\/wiki\/Tetrahedron#Regular_tetrahedron\" target=\"_blank\">Heron-type formula for the volume of a tetrahedron<\/a>)<\/p>\n<p>Fue en el siglo de las luces cuando Lagrange encontr\u00f3 un sencillo modo de calcular el volumen, atendiendo a las coordenadas, que hoy mantenemos. En su art\u00edculo <em>Solutions analytiques de quelques probl\u00e8mes sur les pyramides triangulaires<\/em>, publicado en 1775, propone (las f\u00f3rmulas hoy adaptadas)\u00a0 que el \u00e1rea de un tri\u00e1ngulo es<\/p>\n<p>$$A = \\frac{1}{2!}\\, \\begin{vmatrix} x_1 &amp; y_1 &amp; 1 \\\\ x_2 &amp; y_2\u00a0 &amp; 1 \\\\ x_3 &amp; y_3 &amp; 1 \\end{vmatrix},$$<\/p>\n<p>y el volumen de un tetraedro<\/p>\n<p>$$V = \\frac{1}{3!}\\, \\begin{vmatrix} x_1 &amp; y_1 &amp; z_1 &amp; 1 \\\\ x_2 &amp; y_2 &amp; z_2 &amp; 1 \\\\ x_3 &amp; y_3 &amp; z_3 &amp; 1 \\\\ x_4 &amp; y_4 &amp; z_4 &amp; 1 \\end{vmatrix}$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>El volumen del tetraedro con determinantes.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,9],"tags":[708,206,653],"class_list":["post-4961","post","type-post","status-publish","format-standard","hentry","category-historia","category-personajes","tag-heron-de-alejandria","tag-lagrange","tag-tartaglia","entry"],"_links":{"self":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/4961","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4961"}],"version-history":[{"count":3,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/4961\/revisions"}],"predecessor-version":[{"id":4982,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/4961\/revisions\/4982"}],"wp:attachment":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4961"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4961"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}