{"id":66843,"date":"2026-10-15T09:20:26","date_gmt":"2026-10-15T06:20:26","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66843"},"modified":"2026-10-15T09:20:26","modified_gmt":"2026-10-15T06:20:26","slug":"polygon-interior-angles-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/polygon-interior-angles-calculator\/","title":{"rendered":"Polygon Interior Angles Calculator"},"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\">Find the Interior Angle Sum of a Polygon<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66843\" data-favorite-title=\"Polygon Interior Angles\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/polygon-interior-angles-calculator\/\" data-favorite-excerpt=\"Enter the number of sides to find the sum of interior angles of any polygon. For a regular polygon you also get each interior angle and each exterior angle, with steps shown.\" aria-label=\"Add to Favorites\" 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\/en\/mathematics\/polygon-interior-angles-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66843\" data-post=\"66843\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Number of sides (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=\"number_of_sides_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\">A polygon has at least 3 sides (triangle).<\/small><\/div><div class=\"field is-checkbox\" id=\"input-2\"><input type=\"checkbox\" id=\"field-2\" value=\"1\" checked=\"checked\"\/><label for=\"field-2\">Regular polygon<\/label><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66843\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/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> Your Polygon Angles<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Sum of interior angles<\/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=\"Copy Result\" aria-label=\"Copy Result\"><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>Each interior angle<\/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=\"Copy Result\" aria-label=\"Copy Result\"><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>Each exterior angle<\/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=\"Copy Result\" aria-label=\"Copy Result\"><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>Calculation steps<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><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\">Copy link to this calculation<\/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>Leave a Comment<\/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=\"66843\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/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\">Like <\/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\">Helpful <\/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\">Confused <\/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\">Disappointed <\/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\">Inaccurate <\/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>Polygon Interior Angles Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/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>Where:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 number of sides of the polygon (\\(n \\ge 3\\))<\/li>\n<li>\\(\\Sigma\\alpha\\) \u2014 sum of all interior angles (degrees)<\/li>\n<li>\\(\\alpha\\) \u2014 each interior angle of a <em>regular<\/em> polygon (degrees)<\/li>\n<li>\\(\\beta\\) \u2014 each exterior angle of a <em>regular<\/em> polygon (degrees)<\/li>\n<\/ul>\n<p>The first formula gives the total of all interior angles for any polygon, regular or irregular, based only on how many sides it has. The other two formulas apply to regular polygons, where all angles are equal, so the total is simply divided among the \\(n\\) angles.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.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>Polygon Interior Angles \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a regular hexagon (the calculator's default), which has \\(n = 6\\) equal sides and equal angles.<\/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>Result:<\/strong> the interior angles of a hexagon add up to <strong>720\u00b0<\/strong>; each interior angle of a regular hexagon is <strong>120\u00b0<\/strong>, and each exterior angle is <strong>60\u00b0<\/strong>. Note that \\(120^\\circ + 60^\\circ = 180^\\circ\\), as always for an interior\/exterior angle pair.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Enter the <strong>Number of sides (n)<\/strong> \u2014 any whole number from 3 up, for example 6 for a hexagon.<\/li>\n<li>Leave <strong>Regular polygon<\/strong> ticked if all sides and all angles are equal; untick it for an irregular polygon.<\/li>\n<li>Read the <strong>Sum of interior angles<\/strong> in degrees \u2014 this is valid for every polygon with that number of sides.<\/li>\n<li>When \"Regular polygon\" is ticked, also read <strong>Each interior angle<\/strong> and <strong>Each exterior angle<\/strong>.<\/li>\n<li>Check the <strong>Calculation steps<\/strong> to see the arithmetic written out, with the polygon's name when it has 12 sides or fewer.<\/li>\n<\/ol>\n<h2>What are interior angles of a polygon?<\/h2>\n<p>An interior angle is the angle formed inside the polygon at a vertex, between the two sides that meet there. Every polygon with \\(n\\) sides has exactly \\(n\\) interior angles \u2014 a triangle has 3, a hexagon has 6, and so on.<\/p>\n<p>The key fact is that the <em>sum<\/em> of these angles depends only on the number of sides, not on the polygon's shape. A triangle's angles always total 180\u00b0, whether it is a skinny sliver or perfectly equilateral. Each extra side adds another 180\u00b0 to the total: 180\u00b0 for a triangle, 360\u00b0 for a quadrilateral, 540\u00b0 for a pentagon, 720\u00b0 for a hexagon. That is exactly what the formula \\((n - 2) \\cdot 180^\\circ\\) encodes \u2014 a polygon can be divided into \\(n - 2\\) triangles, each contributing 180\u00b0.<\/p>\n<p>For an <strong>irregular<\/strong> polygon, only the total is fixed; the individual angles can be anything as long as they add up to it. For a <strong>regular<\/strong> polygon \u2014 equal sides and equal angles \u2014 each angle is simply the sum divided by \\(n\\), and each exterior angle is \\(360^\\circ \/ n\\), because the exterior angles must complete a full turn around the polygon.<\/p>\n<p>The result is useful in geometry homework, tiling and design work, and construction. For example, knowing that each interior angle of a regular hexagon is 120\u00b0 tells you why three hexagons meet perfectly around a point (3 \u00d7 120\u00b0 = 360\u00b0), which is the basis of honeycomb patterns.<\/p>\n<h2>Assumptions, limitations and tips<\/h2>\n<ul>\n<li><strong>Convex polygons:<\/strong> the formulas assume a simple polygon. For concave (non-convex) polygons the sum formula still holds, but one or more interior angles exceed 180\u00b0.<\/li>\n<li><strong>Self-intersecting polygons<\/strong> (like a pentagram traced as one path) do not follow these formulas.<\/li>\n<li><strong>Whole numbers only:<\/strong> a polygon cannot have a fractional number of sides, so the calculator asks for a whole number of at least 3.<\/li>\n<li><strong>Exterior angles:<\/strong> the exterior angle shown is for a regular polygon. For any convex polygon, the exterior angles still sum to 360\u00b0, but individually they differ if the polygon is irregular.<\/li>\n<li><strong>Large values:<\/strong> as \\(n\\) grows, each interior angle of a regular polygon approaches 180\u00b0, and each exterior angle approaches 0\u00b0 \u2014 a regular polygon with many sides looks almost like a circle.<\/li>\n<\/ul>\n<p>A handy related fact: the number of diagonals of an \\(n\\)-gon is \\(\\frac{n(n-3)}{2}\\), which comes from choosing 2 vertices out of \\(n\\) \u2014 you can explore that kind of counting with our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/binomial-coefficient\/\">Binomial Coefficient calculator<\/a>. And if you enjoy number-pattern tools, try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/magic-square-generator\/\">Magic Square generator<\/a> as well.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter the number of sides to find the sum of interior angles of any polygon. For a regular polygon you also get each interior angle and each exterior angle, with steps shown.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[23],"class_list":["post-66843","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-geometry"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66843","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66843"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66843\/revisions"}],"predecessor-version":[{"id":67120,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66843\/revisions\/67120"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66843"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66843"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66843"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}