{"id":66833,"date":"2026-10-15T09:20:28","date_gmt":"2026-10-15T06:20:28","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66833"},"modified":"2026-10-15T09:20:28","modified_gmt":"2026-10-15T06:20:28","slug":"arc-length-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/arc-length-calculator\/","title":{"rendered":"Arc Length 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\">Arc Length from Radius and Central Angle<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66833\" data-favorite-title=\"Arc Length\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/arc-length-calculator\/\" data-favorite-excerpt=\"Enter the radius and the central angle of a circle (in degrees, radians or gradians) and this calculator instantly returns the arc length using s = r\u03b8, with the full calculation 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\/arc-length-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66833\" data-post=\"66833\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Radius (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=\"radius_r\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"0\" value=\"5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/option><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/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-1\">Distance from the centre of the circle to the arc.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">Central angle (\u03b8)<\/label><div class=\"term\">\u03b8<\/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=\"number\" name=\"central_angle_\" id=\"field-3\" aria-describedby=\"calc-info-3\" placeholder=\"60\" step=\"any\" value=\"60\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"deg\" selected>deg<\/option><option value=\"rad\">rad<\/option><option value=\"grad\">grad<\/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\">Angle between the two radii that cut off the arc.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66833\">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 Arc Length<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Arc length<\/span><span class=\"result-term\">s<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/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-2\"><span class=\"field-title\"><span>Calculation<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/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=\"66833\" 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>Arc Length 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\\[ s = r \\cdot \\theta \\]\n\\[ \\theta_{\\text{rad}} = \\theta_{\\text{deg}} \\cdot \\frac{\\pi}{180} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(s\\) \u2014 arc length (in the same unit as the radius)<\/li>\n<li>\\(r\\) \u2014 radius of the circle<\/li>\n<li>\\(\\theta\\) \u2014 central angle in <em>radians<\/em> (the angle between the two radii that cut off the arc)<\/li>\n<\/ul>\n<p>The formula says that an arc's length is directly proportional to both the radius and the angle: doubling either one doubles the arc length. If your angle is given in degrees or gradians, it must first be converted to radians, because \\(1\\) radian is defined as the angle that subtends an arc equal in length to the radius.<\/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>Arc Length \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a circle has a radius of 5 m, and you need the length of the arc cut off by a central angle of 60\u00b0.<\/p>\n<p>\\[ \\theta = 60 \\cdot \\frac{\\pi}{180} = \\frac{\\pi}{3} \\approx 1.0472 \\text{ rad} \\]<\/p>\n<p>\\[ s = r \\cdot \\theta = 5 \\times 1.0472 = 5.236 \\text{ m} \\]<\/p>\n<p><strong>Result:<\/strong> the arc length is approximately 5.236 m (exactly \\(\\frac{5\\pi}{3}\\) m, about 17.5% of the full circumference \\(2\\pi r \\approx 31.416\\) m).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Arc Length Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Radius (r)<\/strong> \u2014 the distance from the centre of the circle to the arc \u2014 and pick its unit (m, cm, mm, in or ft).<\/li>\n<li>Enter the <strong>Central angle (\u03b8)<\/strong> between the two radii that cut off the arc, and choose its unit: <code>deg<\/code>, <code>rad<\/code> or <code>grad<\/code>.<\/li>\n<li>Read the <strong>Arc length<\/strong> in the results panel, expressed in the same unit as your radius.<\/li>\n<li>Check the <strong>Calculation<\/strong> line to see the angle converted to radians and the multiplication \\(s = r\\theta\\) performed step by step.<\/li>\n<\/ol>\n<h2>What Is Arc Length?<\/h2>\n<p>An arc is a portion of a circle's circumference between two points. The <em>arc length<\/em> is the distance you would travel if you walked along the circle from one endpoint of the arc to the other \u2014 not the straight-line distance between them (that would be a chord).<\/p>\n<p>The size of an arc is fixed by two quantities only: the radius of the circle and the central angle that subtends it. The central angle is the angle formed at the centre of the circle by the two radii drawn to the arc's endpoints. Because a full circle corresponds to \\(2\\pi\\) radians, an arc covering the whole circle has length \\(2\\pi r\\) \u2014 the familiar circumference formula \u2014 and any smaller angle takes a proportional slice of it.<\/p>\n<p>This is why the angle must be in radians for \\(s = r\\theta\\) to work. A radian is precisely the angle whose arc length equals the radius, so \\(\\theta\\) radians literally means \"\\(\\theta\\) radius-lengths along the curve.\" In degrees, the same relationship requires an extra factor of \\(\\pi\/180\\), which this calculator handles automatically: enter the angle in any of the three supported units and it converts for you.<\/p>\n<p>Arc length appears everywhere a curve is measured: the distance travelled along a circular track, the length of a bent pipe or sheet-metal bend, cable wound on a spool, or the length of a clock hand's tip path over a given time. It is also the foundation of curvilinear motion in physics and of line integrals in calculus.<\/p>\n<h2>Common Central Angles and Their Arcs<\/h2>\n<table>\n<thead>\n<tr>\n<th>Central angle<\/th>\n<th>Fraction of circle<\/th>\n<th>Arc length<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>30\u00b0<\/td>\n<td>1\/12<\/td>\n<td>\\(\\frac{\\pi r}{6} \\approx 0.524\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>45\u00b0<\/td>\n<td>1\/8<\/td>\n<td>\\(\\frac{\\pi r}{4} \\approx 0.785\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>60\u00b0<\/td>\n<td>1\/6<\/td>\n<td>\\(\\frac{\\pi r}{3} \\approx 1.047\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>90\u00b0<\/td>\n<td>1\/4<\/td>\n<td>\\(\\frac{\\pi r}{2} \\approx 1.571\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>180\u00b0<\/td>\n<td>1\/2<\/td>\n<td>\\(\\pi r \\approx 3.142\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>360\u00b0<\/td>\n<td>1<\/td>\n<td>\\(2\\pi r \\approx 6.283\\,r\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note that for angles up to 90\u00b0, the arc is only slightly longer than the radius \u2014 the geometry of radians keeps these numbers intuitive.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Circle only.<\/strong> The formula \\(s = r\\theta\\) applies to circular arcs. For elliptical or arbitrary curves, arc length must be computed with calculus (integration), not with this tool.<\/li>\n<li><strong>Angles larger than a full turn.<\/strong> If you enter more than 360\u00b0 (or \\(2\\pi\\) rad), the calculator still returns \\(r\\theta\\); the arc simply wraps around the circle more than once \u2014 the calculation panel flags this for you.<\/li>\n<li><strong>Same unit in and out.<\/strong> The result is always given in the unit you chose for the radius. Convert beforehand if you need the answer in a different unit.<\/li>\n<li><strong>Sector vs. arc.<\/strong> This tool measures the curved edge only. If you need the area of the pie-slice region bounded by the arc and the two radii, use the sector area formula \\(A = \\tfrac{1}{2}r^2\\theta\\).<\/li>\n<li><strong>Negative angles.<\/strong> A negative angle is treated as an orientation (clockwise), and the arc length comes out negative; enter the absolute value if you only care about the magnitude.<\/li>\n<\/ul>\n<h2>Related Concepts<\/h2>\n<p>Arc length is a classic example of <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/direct-variation\/\">direct variation<\/a>: \\(s\\) varies directly with both \\(r\\) and \\(\\theta\\), so the relationship forms a straight-line proportion. When working through the final multiplication \\(r \\times \\theta_{\\text{rad}}\\) by hand, you can check it with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/multiplication-calculator\/\">multiplication calculator<\/a>, and the degree-to-radian conversion step is a simple <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/division-calculator\/\">division<\/a> by \\(180\/\\pi\\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter the radius and the central angle of a circle (in degrees, radians or gradians) and this calculator instantly returns the arc length using s = r\u03b8, with the full calculation 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-66833","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\/66833","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=66833"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66833\/revisions"}],"predecessor-version":[{"id":67128,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66833\/revisions\/67128"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66833"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66833"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66833"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}