{"id":2173,"date":"2026-10-15T00:41:32","date_gmt":"2026-10-14T23:41:32","guid":{"rendered":"https:\/\/wpcalc.com\/pt\/matematica\/angulos-internos-poligono\/"},"modified":"2026-10-15T00:41:32","modified_gmt":"2026-10-14T23:41:32","slug":"angulos-internos-poligono","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pt\/matematica\/angulos-internos-poligono\/","title":{"rendered":"Calculadora de \u00e2ngulos internos de um pol\u00edgono"},"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\">Descubra a soma dos \u00e2ngulos internos de um pol\u00edgono<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2173\" data-favorite-title=\"\u00c2ngulos internos do pol\u00edgono\" data-favorite-url=\"https:\/\/wpcalc.com\/pt\/matematica\/angulos-internos-poligono\/\" data-favorite-excerpt=\"Digite o n\u00famero de lados para calcular a soma dos \u00e2ngulos internos de qualquer pol\u00edgono. Se ele for regular, voc\u00ea tamb\u00e9m obt\u00e9m cada \u00e2ngulo interno e cada \u00e2ngulo externo, com o passo a passo da resolu\u00e7\u00e3o.\" aria-label=\"Adicionar aos favoritos\" 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\/pt\/matematica\/angulos-internos-poligono\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2173\" data-post=\"2173\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Campos de entrada<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">N\u00famero de lados (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=\"nmero_de_lados_n\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"6\" min=\"3\" value=\"6\"\/><small id=\"calc-info-1\">Um pol\u00edgono tem pelo menos 3 lados (tri\u00e2ngulo).<\/small><\/div><div class=\"field is-checkbox\" id=\"input-2\"><input type=\"checkbox\" id=\"field-2\" value=\"1\" checked=\"checked\"\/><label for=\"field-2\">Pol\u00edgono regular<\/label><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalcular\" id=\"calculate-button\" data-post=\"2173\">Calcular<\/button><button type=\"reset\">Redefinir<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calcular automaticamente<\/label><small>Se ativado, o resultado \u00e9 atualizado automaticamente quando voc\u00ea altera qualquer valor.<\/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> \u00c2ngulos do seu pol\u00edgono<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Soma dos \u00e2ngulos internos<\/span><span class=\"result-term\">\u03a3\u03b1<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"\u00b0\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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>Cada \u00e2ngulo interno<\/span><span class=\"result-term\">\u03b1<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"\u00b0\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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>Cada \u00e2ngulo externo<\/span><span class=\"result-term\">\u03b2<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"\u00b0\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-4\"><span class=\"field-title\"><span>Etapas do c\u00e1lculo<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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\">Copiar link para este c\u00e1lculo<\/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>Deixar um coment\u00e1rio<\/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=\"2173\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">O que voc\u00ea achou deste resultado?<\/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\">Curti <\/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\">\u00datil <\/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\">Confuso <\/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\">Decepcionado <\/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\">Impreciso <\/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>F\u00f3rmula dos \u00e2ngulos internos do pol\u00edgono<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">F\u00f3rmula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\Sigma\\alpha = (n - 2) \\cdot 180^\\circ \\qquad \\alpha = \\frac{(n - 2) \\cdot 180^\\circ}{n} \\qquad \\beta = \\frac{360^\\circ}{n} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Onde:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 n\u00famero de lados do pol\u00edgono (\\(n \\ge 3\\))<\/li>\n<li>\\(\\Sigma\\alpha\\) \u2014 soma de todos os \u00e2ngulos internos (em graus)<\/li>\n<li>\\(\\alpha\\) \u2014 cada \u00e2ngulo interno de um pol\u00edgono <em>regular<\/em> (em graus)<\/li>\n<li>\\(\\beta\\) \u2014 cada \u00e2ngulo externo de um pol\u00edgono <em>regular<\/em> (em graus)<\/li>\n<\/ul>\n<p>A primeira f\u00f3rmula fornece o total de todos os \u00e2ngulos internos de qualquer pol\u00edgono, regular ou n\u00e3o, dependendo apenas do n\u00famero de lados. As outras duas valem para pol\u00edgonos regulares, em que todos os \u00e2ngulos s\u00e3o iguais \u2014 basta dividir o total igualmente entre os \\(n\\) \u00e2ngulos.<\/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>\u00c2ngulos internos do pol\u00edgono \u2013 Exemplo de c\u00e1lculo<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemplo:<\/strong> um hex\u00e1gono regular (o valor padr\u00e3o da calculadora), que tem \\(n = 6\\) lados e \u00e2ngulos iguais.<\/p>\n<p>\\[ \\Sigma\\alpha = (6 - 2) \\cdot 180^\\circ = 4 \\cdot 180^\\circ = 720^\\circ \\]<\/p>\n<p>\\[ \\alpha = \\frac{720^\\circ}{6} = 120^\\circ \\]<\/p>\n<p>\\[ \\beta = \\frac{360^\\circ}{6} = 60^\\circ \\]<\/p>\n<p><strong>Resultado:<\/strong> os \u00e2ngulos internos de um hex\u00e1gono somam <strong>720\u00b0<\/strong>; cada \u00e2ngulo interno de um hex\u00e1gono regular mede <strong>120\u00b0<\/strong>, e cada \u00e2ngulo externo, <strong>60\u00b0<\/strong>. Note que \\(120^\\circ + 60^\\circ = 180^\\circ\\), como sempre acontece com um par de \u00e2ngulos interno e externo.<\/p>\n<p>\n\r\n<\/section>\n<h2>Como usar a calculadora<\/h2>\n<ol>\n<li>Digite o <strong>N\u00famero de lados (n)<\/strong> \u2014 qualquer n\u00famero inteiro a partir de 3, por exemplo 6 para um hex\u00e1gono.<\/li>\n<li>Mantenha a op\u00e7\u00e3o <strong>Pol\u00edgono regular<\/strong> marcada se todos os lados e todos os \u00e2ngulos forem iguais; desmarque-a para um pol\u00edgono irregular.<\/li>\n<li>Leia a <strong>Soma dos \u00e2ngulos internos<\/strong> em graus \u2014 ela vale para qualquer pol\u00edgono com esse n\u00famero de lados.<\/li>\n<li>Com \"Pol\u00edgono regular\" marcado, voc\u00ea tamb\u00e9m v\u00ea <strong>Cada \u00e2ngulo interno<\/strong> e <strong>Cada \u00e2ngulo externo<\/strong>.<\/li>\n<li>Confira os <strong>Passos do c\u00e1lculo<\/strong> para ver todas as contas escritas, junto com o nome do pol\u00edgono quando ele tem 12 lados ou menos.<\/li>\n<\/ol>\n<h2>O que s\u00e3o os \u00e2ngulos internos de um pol\u00edgono?<\/h2>\n<p>Um \u00e2ngulo interno \u00e9 o \u00e2ngulo formado dentro do pol\u00edgono, em um v\u00e9rtice, entre os dois lados que se encontram ali. Todo pol\u00edgono com \\(n\\) lados tem exatamente \\(n\\) \u00e2ngulos internos \u2014 um tri\u00e2ngulo tem 3, um hex\u00e1gono tem 6, e assim por diante.<\/p>\n<p>O ponto essencial \u00e9 que a <em>soma<\/em> desses \u00e2ngulos depende apenas do n\u00famero de lados, e n\u00e3o do formato do pol\u00edgono. Os \u00e2ngulos de um tri\u00e2ngulo sempre somam 180\u00b0, seja ele bem fino ou perfeitamente equil\u00e1tero. Cada lado a mais acrescenta 180\u00b0 ao total: 180\u00b0 para o tri\u00e2ngulo, 360\u00b0 para o quadril\u00e1tero, 540\u00b0 para o pent\u00e1gono, 720\u00b0 para o hex\u00e1gono. \u00c9 exatamente isso que a f\u00f3rmula \\((n - 2) \\cdot 180^\\circ\\) expressa \u2014 um pol\u00edgono pode ser dividido em \\(n - 2\\) tri\u00e2ngulos, e cada tri\u00e2ngulo contribui com 180\u00b0.<\/p>\n<p>Em um pol\u00edgono <strong>irregular<\/strong>, apenas o total \u00e9 fixo; os \u00e2ngulos individuais podem ter qualquer valor, desde que a soma continue a mesma. J\u00e1 em um pol\u00edgono <strong>regular<\/strong> \u2014 lados e \u00e2ngulos iguais \u2014 cada \u00e2ngulo \u00e9 simplesmente a soma dividida por \\(n\\), e cada \u00e2ngulo externo vale \\(360^\\circ \/ n\\), pois os \u00e2ngulos externos precisam completar uma volta inteira ao redor do pol\u00edgono.<\/p>\n<p>Esse resultado \u00e9 \u00fatil em exerc\u00edcios de geometria, em ladrilhamento e design e na constru\u00e7\u00e3o civil. Por exemplo, saber que cada \u00e2ngulo interno de um hex\u00e1gono regular mede 120\u00b0 explica por que tr\u00eas hex\u00e1gonos se encaixam perfeitamente em torno de um ponto (3 \u00d7 120\u00b0 = 360\u00b0) \u2014 a base dos padr\u00f5es em formato de colmeia.<\/p>\n<h2>Premissas, limita\u00e7\u00f5es e dicas<\/h2>\n<ul>\n<li><strong>Pol\u00edgonos convexos:<\/strong> as f\u00f3rmulas pressup\u00f5em um pol\u00edgono simples. Em pol\u00edgonos c\u00f4ncavos (n\u00e3o convexos), a f\u00f3rmula da soma continua v\u00e1lida, mas um ou mais \u00e2ngulos internos passam de 180\u00b0.<\/li>\n<li><strong>Pol\u00edgonos que se autointersectam<\/strong> (como um pentagrama tra\u00e7ado de uma s\u00f3 vez) n\u00e3o seguem essas f\u00f3rmulas.<\/li>\n<li><strong>Apenas n\u00fameros inteiros:<\/strong> um pol\u00edgono n\u00e3o pode ter um n\u00famero fracion\u00e1rio de lados, por isso a calculadora pede um n\u00famero inteiro maior ou igual a 3.<\/li>\n<li><strong>\u00c2ngulos externos:<\/strong> o \u00e2ngulo externo exibido vale para um pol\u00edgono regular. Em qualquer pol\u00edgono convexo, os \u00e2ngulos externos continuam somando 360\u00b0, mas, individualmente, s\u00e3o diferentes se o pol\u00edgono for irregular.<\/li>\n<li><strong>Valores grandes:<\/strong> \u00e0 medida que \\(n\\) cresce, cada \u00e2ngulo interno de um pol\u00edgono regular se aproxima de 180\u00b0, e cada \u00e2ngulo externo se aproxima de 0\u00b0 \u2014 um pol\u00edgono regular com muitos lados parece quase um c\u00edrculo.<\/li>\n<\/ul>\n<p>Um fato relacionado que vale a pena conhecer: o n\u00famero de diagonais de um pol\u00edgono de \\(n\\) lados \u00e9 \\(\\frac{n(n-3)}{2}\\), que vem de escolher 2 v\u00e9rtices entre os \\(n\\) \u2014 voc\u00ea pode explorar esse tipo de contagem com nossa <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/coeficiente-binomial\/\">calculadora de Coeficiente Binomial<\/a>. E se voc\u00ea gosta de ferramentas com padr\u00f5es num\u00e9ricos, experimente tamb\u00e9m o <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/gerador-de-quadrado-magico\/\">gerador de Quadrado M\u00e1gico<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Digite o n\u00famero de lados para calcular a soma dos \u00e2ngulos internos de qualquer pol\u00edgono. Se ele for regular, voc\u00ea tamb\u00e9m obt\u00e9m cada \u00e2ngulo interno e cada \u00e2ngulo externo, com o passo a passo da resolu\u00e7\u00e3o.<\/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":[8],"tags":[42],"class_list":["post-2173","post","type-post","status-publish","format-standard","hentry","category-matematica","tag-geometria"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/comments?post=2173"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2173\/revisions"}],"predecessor-version":[{"id":2247,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2173\/revisions\/2247"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/media?parent=2173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/categories?post=2173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/tags?post=2173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}