{"id":5824,"date":"2026-10-15T03:34:35","date_gmt":"2026-10-15T00:34:35","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/mathematiques\/aire-polygone-regulier\/"},"modified":"2026-10-15T03:34:35","modified_gmt":"2026-10-15T00:34:35","slug":"aire-polygone-regulier","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/mathematiques\/aire-polygone-regulier\/","title":{"rendered":"Calculateur d'aire d'un polygone r\u00e9gulier"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Calculer l&#039;aire d&#039;un polygone r\u00e9gulier<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"5824\" data-favorite-title=\"Aire d&#039;un polygone r\u00e9gulier\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/mathematiques\/aire-polygone-regulier\/\" data-favorite-excerpt=\"Calculez l&#039;aire de n&#039;importe quel polygone r\u00e9gulier \u00e0 partir de la longueur d&#039;un c\u00f4t\u00e9, de l&#039;apoth\u00e8me ou du rayon du cercle circonscrit. Saisissez le nombre de c\u00f4t\u00e9s et une dimension connue pour obtenir l&#039;aire, le p\u00e9rim\u00e8tre et les angles.\" aria-label=\"Ajouter aux favoris\" data-favorite-icon=\"icon icon-geometry\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/fr\/mathematiques\/aire-polygone-regulier\/\" method=\"POST\" class=\"calculator\" id=\"calculator-5824\" data-post=\"5824\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Nombre de c\u00f4t\u00e9s (n)<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"nombre_de_cts_n\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"6\" min=\"3\" max=\"10000\" value=\"6\"\/><small id=\"calc-info-1\">Nombre entier de c\u00f4t\u00e9s \u00e9gaux, 3 ou plus.<\/small><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Donn\u00e9e connue<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"donne_connue\"><option value=\"side\" selected>Longueur de c\u00f4t\u00e9 (s)<\/option><option value=\"apothem\">Apoth\u00e8me (a)<\/option><option value=\"circumradius\">Rayon circonscrit (R)<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Longueur de c\u00f4t\u00e9 (s)<\/label><div class=\"term\">s<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"longueur_de_ct_s\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"10\" min=\"0\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\u00e9\"><option value=\"m\">m<\/option><option value=\"cm\" selected>cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-3\">Longueur d&#039;un c\u00f4t\u00e9 du polygone.<\/small><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-5\">Apoth\u00e8me (a)<\/label><div class=\"term\">a<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"apothme_a\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"8.66\" min=\"0\" value=\"8.66\"\/><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\">m<\/option><option value=\"cm\" selected>cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-5\">Distance du centre au milieu d&#039;un c\u00f4t\u00e9 (rayon du cercle inscrit).<\/small><\/div><div class=\"field has-term\" id=\"input-5\" hidden><label for=\"field-7\">Rayon circonscrit (R)<\/label><div class=\"term\">R<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"rayon_circonscrit_r\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"10\" min=\"0\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\">m<\/option><option value=\"cm\" selected>cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-7\">Distance du centre \u00e0 un sommet (rayon du cercle circonscrit).<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalculer\" id=\"calculate-button\" data-post=\"5824\">Calculer<\/button><button type=\"reset\">R\u00e9initialiser<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculer automatiquement<\/label><small>Si activ\u00e9, le r\u00e9sultat se met \u00e0 jour automatiquement d\u00e8s que vous modifiez une valeur.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Votre polygone<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Aire<\/span><span class=\"result-term\">A<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>P\u00e9rim\u00e8tre<\/span><span class=\"result-term\">P<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Longueur de c\u00f4t\u00e9<\/span><span class=\"result-term\">s<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Apoth\u00e8me<\/span><span class=\"result-term\">a<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Rayon circonscrit<\/span><span class=\"result-term\">R<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-6\"><span class=\"field-title\"><span>Angle int\u00e9rieur<\/span><span class=\"result-term\">\u03b1<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-7\"><span class=\"field-title\"><span>\u00c9tapes de calcul<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copier le lien vers ce calcul<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Laisser un commentaire<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"5824\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Que pensez-vous de ce r\u00e9sultat\u00a0?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">J\u2019aime <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Utile <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confus <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">D\u00e9\u00e7u <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inexact <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Aire d'un polygone r\u00e9gulier<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formule<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ A = \\frac{n\\,s^2}{4\\tan(\\pi\/n)} = \\frac{1}{2}\\,P\\,a = \\frac{1}{2}\\,n\\,R^2\\sin\\!\\left(\\frac{2\\pi}{n}\\right) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>O\u00f9&nbsp;:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 nombre de c\u00f4t\u00e9s (\\(n \\ge 3\\))<\/li>\n<li>\\(s\\) \u2014 longueur d'un c\u00f4t\u00e9<\/li>\n<li>\\(a\\) \u2014 apoth\u00e8me (distance du centre au milieu d'un c\u00f4t\u00e9, c'est-\u00e0-dire le rayon du cercle inscrit)<\/li>\n<li>\\(R\\) \u2014 rayon du cercle circonscrit (distance du centre \u00e0 un sommet)<\/li>\n<li>\\(P = n \\cdot s\\) \u2014 p\u00e9rim\u00e8tre<\/li>\n<li>\\(\\pi \\approx 3{,}14159\\)<\/li>\n<\/ul>\n<p>Ces trois expressions donnent la m\u00eame aire&nbsp;; utilisez celle qui correspond \u00e0 la dimension que vous connaissez. La forme centrale, \\(A = \\tfrac{1}{2}Pa\\), est le r\u00e9sultat classique obtenu en d\u00e9composant le polygone en \\(n\\) triangles identiques, chacun ayant pour base \\(s\\) et pour hauteur \\(a\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Aire d'un polygone r\u00e9gulier \u2013 Exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple&nbsp;:<\/strong> un hexagone r\u00e9gulier dont le c\u00f4t\u00e9 mesure 10 cm.<\/p>\n<p>\\[ a = \\frac{s}{2\\tan(\\pi\/n)} = \\frac{10}{2\\tan(\\pi\/6)} = \\frac{10}{2 \\cdot 0{,}5774} \\approx 8{,}660 \\text{ cm} \\]<\/p>\n<p>\\[ P = n \\cdot s = 6 \\cdot 10 = 60 \\text{ cm} \\]<\/p>\n<p>\\[ A = \\tfrac{1}{2}\\,P\\,a = \\tfrac{1}{2} \\cdot 60 \\cdot 8{,}660 \\approx 259{,}81 \\text{ cm}^2 \\]<\/p>\n<p><strong>R\u00e9sultat&nbsp;:<\/strong> l'aire est d'environ <strong>259,81 cm\u00b2<\/strong>. Le calculateur indique \u00e9galement un rayon de cercle circonscrit \\(R = 10\\) cm et un angle int\u00e9rieur de 120\u00b0.<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur<\/h2>\n<ol>\n<li>Saisissez le <strong>Nombre de c\u00f4t\u00e9s (n)<\/strong> \u2014 n'importe quel nombre entier de 3 (triangle) jusqu'\u00e0 10 000.<\/li>\n<li>Choisissez la dimension que vous connaissez dans le menu d\u00e9roulant <strong>Grandeur connue<\/strong>&nbsp;: <em>Longueur du c\u00f4t\u00e9 (s)<\/em>, <em>Apoth\u00e8me (a)<\/em> ou <em>Rayon du cercle circonscrit (R)<\/em>. Seul le champ de saisie correspondant reste visible.<\/li>\n<li>Saisissez la valeur dans le champ qui appara\u00eet \u2014 <strong>Longueur du c\u00f4t\u00e9 (s)<\/strong>, <strong>Apoth\u00e8me (a)<\/strong> ou <strong>Rayon du cercle circonscrit (R)<\/strong> \u2014 et choisissez son unit\u00e9 (m, cm, mm, ft ou in).<\/li>\n<li>Lisez les r\u00e9sultats&nbsp;: l'<strong>Aire<\/strong> (dans l'unit\u00e9 carr\u00e9e correspondante), ainsi que le <strong>P\u00e9rim\u00e8tre<\/strong>, la <strong>Longueur du c\u00f4t\u00e9<\/strong>, l'<strong>Apoth\u00e8me<\/strong>, le <strong>Rayon du cercle circonscrit<\/strong>, l'<strong>Angle int\u00e9rieur<\/strong> et le d\u00e9roul\u00e9 du calcul \u00e9tape par \u00e9tape.<\/li>\n<\/ol>\n<h2>Qu'est-ce qu'un polygone r\u00e9gulier ?<\/h2>\n<p>Un polygone r\u00e9gulier est une figure ferm\u00e9e \u00e0 c\u00f4t\u00e9s rectilignes dont tous les c\u00f4t\u00e9s ont la m\u00eame longueur et tous les angles int\u00e9rieurs sont \u00e9gaux. Parmi les exemples courants figurent le triangle \u00e9quilat\u00e9ral (\\(n=3\\)), le carr\u00e9 (\\(n=4\\)), le pentagone r\u00e9gulier (\\(n=5\\)) et l'hexagone r\u00e9gulier (\\(n=6\\)). Comme tous les c\u00f4t\u00e9s et tous les angles concordent, un polygone r\u00e9gulier est enti\u00e8rement d\u00e9termin\u00e9 par seulement deux nombres&nbsp;: le nombre de c\u00f4t\u00e9s et une dimension lin\u00e9aire.<\/p>\n<p>La formule de l'aire consiste \u00e0 d\u00e9couper le polygone en \\(n\\) triangles superposables qui se rejoignent au centre. Chaque triangle a pour base \\(s\\) (un c\u00f4t\u00e9 du polygone) et pour hauteur \\(a\\) (l'apoth\u00e8me), donc son aire vaut \\(\\tfrac{1}{2}sa\\). En multipliant par les \\(n\\) c\u00f4t\u00e9s, on obtient \\(A = \\tfrac{1}{2}na s = \\tfrac{1}{2}Pa\\). C'est la m\u00eame logique qui permet de retrouver l'aire du cercle \u00e0 la limite : quand \\(n\\) devient tr\u00e8s grand, le polygone devient indiscernable d'un cercle de rayon \\(R\\) et la formule tend vers \\(\\pi R^2\\).<\/p>\n<p>Deux dimensions d\u00e9riv\u00e9es m\u00e9ritent d'\u00eatre comprises. L'<strong>apoth\u00e8me<\/strong> est le rayon du cercle inscrit \u2014 le plus grand cercle que l'on peut loger dans le polygone, tangent \u00e0 chaque c\u00f4t\u00e9 en son milieu. Le <strong>rayon du cercle circonscrit<\/strong> est le rayon du cercle qui passe par tous les sommets. Les trois dimensions sont reli\u00e9es par les identit\u00e9s \\(s = 2a\\tan(\\pi\/n) = 2R\\sin(\\pi\/n)\\) et \\(a = R\\cos(\\pi\/n)\\), c'est pourquoi le calculateur peut toutes les d\u00e9duire \u00e0 partir d'une seule.<\/p>\n<p>L'<strong>angle int\u00e9rieur<\/strong> affich\u00e9 dans les r\u00e9sultats provient de la formule \\(\\alpha = (n-2) \\cdot 180\u00b0\/n\\). Il est utile en construction, en carrelage et en menuiserie&nbsp;: par exemple, un hexagone a des angles de 120\u00b0, il faut donc couper les planches \u00e0 60\u00b0 en onglet pour les assembler.<\/p>\n<h2>Hypoth\u00e8ses, limites et conseils<\/h2>\n<ul>\n<li><strong>Polygones r\u00e9guliers uniquement.<\/strong> Les formules supposent que tous les c\u00f4t\u00e9s et tous les angles sont \u00e9gaux. Pour un polygone irr\u00e9gulier, d\u00e9composez-le en triangles ou en trap\u00e8zes et additionnez les aires.<\/li>\n<li><strong>Unit\u00e9s coh\u00e9rentes obligatoires.<\/strong> Le menu des unit\u00e9s convertit automatiquement la saisie et l'aire est renvoy\u00e9e dans l'unit\u00e9 carr\u00e9e correspondante (par ex. cm \u2192 cm\u00b2). Si vous calculez \u00e0 la main, exprimez toutes les longueurs dans la m\u00eame unit\u00e9 avant d'appliquer la formule.<\/li>\n<li><strong>Choisissez la dimension que vous avez r\u00e9ellement mesur\u00e9e.<\/strong> En pratique, on conna\u00eet souvent l'entre-flats (le double de l'apoth\u00e8me, pour \\(n\\) pair) ou le rayon du cercle des trous de fixation (rayon circonscrit) plut\u00f4t qu'un c\u00f4t\u00e9 \u2014 s\u00e9lectionnez cette option pour \u00e9viter les erreurs d'arrondi dues \u00e0 une conversion interm\u00e9diaire.<\/li>\n<li><strong>Valeurs \u00e9lev\u00e9es de \\(n\\).<\/strong> Les valeurs vont jusqu'\u00e0 10 000 ; pour de telles figures, le r\u00e9sultat correspond pratiquement \u00e0 l'aire d'un cercle de rayon \\(R\\), puisque \\(A \\to \\pi R^2\\) quand \\(n \\to \\infty\\).<\/li>\n<li><strong>Arrondi.<\/strong> Les r\u00e9sultats sont affich\u00e9s avec trois d\u00e9cimales. Les valeurs interm\u00e9diaires des \u00e9tapes de calcul sont arrondies pour l'affichage : une v\u00e9rification \u00e0 la main peut donc diff\u00e9rer sur le dernier chiffre.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculez l'aire de n'importe quel polygone r\u00e9gulier \u00e0 partir de la longueur d'un c\u00f4t\u00e9, de l'apoth\u00e8me ou du rayon du cercle circonscrit. Saisissez le nombre de c\u00f4t\u00e9s et une dimension connue pour obtenir l'aire, le p\u00e9rim\u00e8tre et les angles.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[3],"class_list":["post-5824","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-geometrie"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/5824","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/comments?post=5824"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/5824\/revisions"}],"predecessor-version":[{"id":5863,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/5824\/revisions\/5863"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=5824"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=5824"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=5824"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}