{"id":987,"date":"2016-10-07T21:14:39","date_gmt":"2016-10-07T19:14:39","guid":{"rendered":"http:\/\/www.francou.xyz\/?page_id=987"},"modified":"2025-11-11T11:46:33","modified_gmt":"2025-11-11T10:46:33","slug":"racine-carree-de-3-2","status":"publish","type":"page","link":"https:\/\/www.tokotidi.net\/?page_id=987","title":{"rendered":"Racine carr\u00e9e de 3"},"content":{"rendered":"\n<p>1,7320508075688774<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"260\" height=\"276\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/star-made-of-twelve-equilateral-triangles.png\" alt=\"\" class=\"wp-image-1227\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"450\" height=\"280\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/fibonacci-spiral.gif\" alt=\"\" class=\"wp-image-1225\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"286\" height=\"176\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/fibonacci-spiral-1.png\" alt=\"\" class=\"wp-image-1221\"\/><\/figure>\n\n\n\n<p>Si on encadre un cercle de diam\u00e8tre 1 avec un triangle \u00e9quilat\u00e9ral, chaque c\u00f4t\u00e9 du triangle est \u00e9gal \u00e0 racine carr\u00e9e de 3. Et donc chaque carr\u00e9 \u00e0 une aire de 3 :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"491\" height=\"450\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/flat-prism.jpg\" alt=\"\" class=\"wp-image-1218\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/flat-prism.jpg 491w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/flat-prism-300x275.jpg 300w\" sizes=\"auto, (max-width: 491px) 100vw, 491px\" \/><\/figure>\n\n\n\n<p>L&rsquo;aire du triangle \u00e9quilat\u00e9ral est ( cot\u00e9 \u00b2 x racine de 3 ) \/ 4. Donc dans ce cas : ( 3 x racine de 3)  \/4  = 1,2990381&#8230;<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>L&rsquo;aire d&rsquo;une sph\u00e8re est \u00e9gale \u00e0 4 fois l&rsquo;aire du disque de m\u00eame diam\u00e8tre :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"225\" height=\"225\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/sphere-great-circle.jpg\" alt=\"\" class=\"wp-image-1216\" style=\"width:226px;height:auto\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/sphere-great-circle.jpg 225w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/sphere-great-circle-150x150.jpg 150w\" sizes=\"auto, (max-width: 225px) 100vw, 225px\" \/><\/figure>\n\n\n\n<p>ou on peut dire aussi que l&rsquo;aire d&rsquo;une demi-sph\u00e8re est \u00e9gale \u00e0 2 fois l&rsquo;aire du disque :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"311\" src=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aire-disque-et-aire-demi-sphere.jpeg\" alt=\"\" class=\"wp-image-1489\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aire-disque-et-aire-demi-sphere.jpeg 600w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aire-disque-et-aire-demi-sphere-300x156.jpeg 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"225\" height=\"224\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/spiral-of-archimedes.jpg\" alt=\"\" class=\"wp-image-1214\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/spiral-of-archimedes.jpg 225w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/spiral-of-archimedes-150x150.jpg 150w\" sizes=\"auto, (max-width: 225px) 100vw, 225px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"218\" height=\"231\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/Circumscribed-cylinder-to-a-sphere.jpg\" alt=\"\" class=\"wp-image-1212\"\/><\/figure>\n\n\n\n<p>Les Surfaces (aires) de la sph\u00e8re et du cylindre qui l&rsquo;entoure sont \u00e9gales tranche \u00e0 tranche (en bleu plus clair sur les sch\u00e9mas ci-dessous) : <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"436\" height=\"145\" src=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aires-sphere-et-et-cylindre.png\" alt=\"\" class=\"wp-image-1483\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aires-sphere-et-et-cylindre.png 436w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aires-sphere-et-et-cylindre-300x100.png 300w\" sizes=\"auto, (max-width: 436px) 100vw, 436px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"601\" height=\"287\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/peripheral-surface-sphere-cylinder.png\" alt=\"\" class=\"wp-image-1209\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/peripheral-surface-sphere-cylinder.png 601w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/peripheral-surface-sphere-cylinder-300x143.png 300w\" sizes=\"auto, (max-width: 601px) 100vw, 601px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"727\" height=\"319\" src=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aires-sphere-et-et-cylindre-tranche-a-tranche.png\" alt=\"\" class=\"wp-image-1481\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aires-sphere-et-et-cylindre-tranche-a-tranche.png 727w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2025\/11\/Aires-sphere-et-et-cylindre-tranche-a-tranche-300x132.png 300w\" sizes=\"auto, (max-width: 727px) 100vw, 727px\" \/><\/figure>\n\n\n\n<p>L&rsquo;aire du cylindre est un rectangle referm\u00e9 sur lui-m\u00eame.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Proportions entre un c\u00f4ne, une sph\u00e8re, un cylindre<\/h2>\n\n\n\n<p>Si on part d&rsquo;un cercle de diam\u00e8tre 1 :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"350\" height=\"175\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/350px-Inscribed_cone_sphere_cylinder.svg_.png\" alt=\"\" class=\"wp-image-1206\" style=\"width:560px;height:auto\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/350px-Inscribed_cone_sphere_cylinder.svg_.png 350w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/350px-Inscribed_cone_sphere_cylinder.svg_-300x150.png 300w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/figure>\n\n\n\n<p>Le volume du c\u00f4ne de diam\u00e8tre 1 et de hauteur 1 est \u00e9gal \u00e0 : (\u03c0\u00d7(0.5)<sup>2<\/sup>\u00d71) \/ 3 =<strong>0.26<\/strong><\/p>\n\n\n\n<p>Le volume de la sph\u00e8re de diam\u00e8tre 1 est \u00e9gal \u00e0 : (4\u00d7\u03c0\u00d70.5<sup>3<\/sup>) \/ 3=<strong>0.52, soit deux fois le volume du c\u00f4ne<\/strong><\/p>\n\n\n\n<p>Le volume du cylindre est \u00e9gal \u00e0 : \u03c0 \u00d7 0,5<sup>2<\/sup> \u00d7 1 = <strong>0,7854, soit une fois et demi le volume de la sph\u00e8re, ou 3 fois le volume du c\u00f4ne.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"407\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/square-roots-spiral-suite_2.png\" alt=\"\" class=\"wp-image-1202\" style=\"width:840px;height:auto\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/square-roots-spiral-suite_2.png 500w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/square-roots-spiral-suite_2-300x244.png 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>On retrouve la racine de 3 dans la g\u00e9om\u00e9trie du cube. Si on a un cube avec un c\u00f4t\u00e9 de 1, alors la diagonale du cube est racine de 3 : <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"244\" height=\"215\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/one-unit-cube-square-root-of-3-diagonal.jpg\" alt=\"\" class=\"wp-image-1200\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>Si on consid\u00e8re un triangle \u00e9quilat\u00e9ral de 1 de c\u00f4t\u00e9, son aire est de (racine de 3) \/ 4:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"107\" height=\"54\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/equilateral-triangle-area.png\" alt=\"\" class=\"wp-image-1197\"\/><\/figure>\n\n\n\n<p>Et l&rsquo;aire du cercle circonscrit est de (racine de 3) \/ 3<\/p>\n\n\n\n<p>Et l&rsquo;aire du disque inscrit est de (racine de 3) \/ 6 <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"686\" height=\"585\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/incircle-and-circumcircle-of-an-equilateral-triangle.png\" alt=\"\" class=\"wp-image-1195\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/incircle-and-circumcircle-of-an-equilateral-triangle.png 686w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/incircle-and-circumcircle-of-an-equilateral-triangle-300x256.png 300w\" sizes=\"auto, (max-width: 686px) 100vw, 686px\" \/><\/figure>\n\n\n\n<p>On peut alors se demander \u00e0 quoi correspondrait l&rsquo;aire (racine de 3) \/ 5 , dont la valeur est 0,34641016<\/p>\n\n\n\n<p>Dans cette figure int\u00e9ressante, on peut aussi noter que l&rsquo;aire de la zone rose est \u00e9gale \u00e0 3 fois l&rsquo;aire de la zone jaune :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/PairsOfAreas3.gif\" alt=\"\" class=\"wp-image-1193\"\/><\/figure>\n\n\n\n<p>Et donc l&rsquo;aire de chacun des sous-ensembles roses est \u00e9gale \u00e0 l&rsquo;aire du disque jaune : <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"446\" height=\"447\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/PairsOfAreas2.gif\" alt=\"\" class=\"wp-image-1190\"\/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"446\" height=\"447\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/PairsOfAreas.gif\" alt=\"\" class=\"wp-image-1189\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"272\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/square-root-circle-tau.png\" alt=\"\" class=\"wp-image-1187\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"584\" height=\"529\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/square-root-circle.png\" alt=\"\" class=\"wp-image-1185\" srcset=\"https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/square-root-circle.png 584w, https:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/square-root-circle-300x272.png 300w\" sizes=\"auto, (max-width: 584px) 100vw, 584px\" \/><\/figure>\n\n\n\n<p>Et la racine de 3 dans un hexagone. Si la largeur de l&rsquo;hexagone est 2, alors sa hauteur est racine de 3 :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"220\" height=\"190\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/220px-Root_3_Hexagon_svg3.png\" alt=\"\" class=\"wp-image-1183\"\/><\/figure>\n\n\n\n<p>Et dans l&rsquo;aire d&rsquo;un triangle \u00e9quilat\u00e9ral. Si le c\u00f4t\u00e9 fait 1, alors l&rsquo;aire est \u00e9gale \u00e0 (racine carr\u00e9e de 3) \/ 4 :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"211\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/area-equilateral-triangle.gif\" alt=\"\" class=\"wp-image-1181\"\/><\/figure>\n\n\n\n<p>Et si le c\u00f4t\u00e9 fait 2, alors la hauteur du triangle est racine carr\u00e9e de 3 :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"184\" height=\"196\" src=\"http:\/\/www.tokotidi.net\/wp-content\/uploads\/2020\/10\/184px-30-60-90_svg.png\" alt=\"\" class=\"wp-image-1179\"\/><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>1,7320508075688774 Si on encadre un cercle de diam\u00e8tre 1 avec un triangle \u00e9quilat\u00e9ral, chaque c\u00f4t\u00e9 du triangle est \u00e9gal \u00e0 racine carr\u00e9e de 3. Et donc chaque carr\u00e9 \u00e0 une aire de 3 : L&rsquo;aire du triangle \u00e9quilat\u00e9ral est ( cot\u00e9 \u00b2 x racine de 3 ) \/ 4. Donc dans ce cas : ( [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-987","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=\/wp\/v2\/pages\/987","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=987"}],"version-history":[{"count":153,"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=\/wp\/v2\/pages\/987\/revisions"}],"predecessor-version":[{"id":1497,"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=\/wp\/v2\/pages\/987\/revisions\/1497"}],"wp:attachment":[{"href":"https:\/\/www.tokotidi.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=987"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}