{"id":61919,"date":"2021-05-27T11:15:30","date_gmt":"2021-05-27T09:15:30","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61919"},"modified":"2026-09-30T20:36:41","modified_gmt":"2026-09-30T17:36:41","slug":"length-angle-bisector-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/length-angle-bisector-calculator\/","title":{"rendered":"Length of the Angle Bisector 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\">Angle Bisector Length Calculator<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61919\" data-favorite-title=\"Length of the Angle Bisector\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/length-angle-bisector-calculator\/\" data-favorite-excerpt=\"Enter the three side lengths of a triangle to find the length of the internal angle bisector \u2014 the segment that splits the angle between Side 1 and Side 2 into two equal parts.\" 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\/length-angle-bisector-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61919\" data-post=\"61919\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Side 1<\/label><input type=\"text\" name=\"side_1\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"0\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Side 2<\/label><input type=\"text\" name=\"side_2\" id=\"field-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"7\" min=\"0\" value=\"7\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Side 3<\/label><input type=\"text\" name=\"side_3\" id=\"field-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"6\" min=\"0\" value=\"6\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61919\">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 Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Length of angle bisector<\/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><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=\"61919\" 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>Angle Bisector 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\\[ l = \\sqrt{ab\\left(1 - \\frac{c^{2}}{(a+b)^{2}}\\right)} = \\frac{\\sqrt{ab\\left((a+b)^{2} - c^{2}\\right)}}{a+b} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(l\\) \u2014 length of the bisector of the angle between sides \\(a\\) and \\(b\\)<\/li>\n<li>\\(a\\) and \\(b\\) \u2014 the two sides of the triangle that form the bisected angle (Side 1 and Side 2 in the calculator)<\/li>\n<li>\\(c\\) \u2014 the side opposite the bisected angle (Side 3 in the calculator)<\/li>\n<\/ul>\n<p>An equivalent form is \\( l = \\dfrac{2ab\\cos(C\/2)}{a+b} \\), where \\(C\\) is the bisected angle. Both expressions give the length of the internal angle bisector measured from the vertex to the opposite side.<\/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>Angle Bisector Length \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a triangle has sides \\(a = 5\\), \\(b = 7\\), and \\(c = 6\\). Find the length of the bisector of the angle between sides \\(a\\) and \\(b\\).<\/p>\n<p>\\[ l = \\sqrt{5 \\cdot 7 \\left(1 - \\frac{6^{2}}{(5+7)^{2}}\\right)} = \\sqrt{35\\left(1 - \\frac{36}{144}\\right)} = \\sqrt{35 \\cdot 0.75} = \\sqrt{26.25} \\approx 5.123 \\]<\/p>\n<p><strong>Result:<\/strong> the angle bisector has a length of approximately <strong>5.123<\/strong> units.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the length of the first side forming the bisected angle in the <strong>Side 1<\/strong> field.<\/li>\n<li>Enter the length of the second side forming that angle in the <strong>Side 2<\/strong> field.<\/li>\n<li>Enter the length of the side opposite the angle in the <strong>Side 3<\/strong> field.<\/li>\n<li>The calculator instantly displays the <strong>Length of angle bisector<\/strong> \u2014 the segment from the vertex between Side 1 and Side 2 to the opposite side, rounded to three decimal places.<\/li>\n<\/ol>\n<h2>What Is an Angle Bisector?<\/h2>\n<p>An angle bisector of a triangle is a line segment that starts at a vertex, splits that vertex's angle into two equal angles, and ends on the opposite side. Every triangle has three internal angle bisectors, one from each vertex. They all meet at a single point called the incenter, which is the center of the triangle's inscribed circle.<\/p>\n<p>The bisector has a useful property: it divides the opposite side into segments proportional to the two adjacent sides. If the bisector from the angle between sides \\(a\\) and \\(b\\) meets side \\(c\\), then the two segments of \\(c\\) are in the ratio \\(a : b\\). This property, together with the Law of Cosines or Stewart's theorem, leads directly to the length formula used above.<\/p>\n<p>Knowing the bisector length matters in many practical situations: constructing triangles in geometry class, verifying symmetry in architectural and engineering drawings, computing areas by splitting a triangle into two smaller ones, or positioning elements in CAD models. Because the formula needs only the three side lengths, no angles have to be measured separately.<\/p>\n<p>Note that the result is always shorter than both adjacent sides, and it can never exceed half the sum \\(a + b\\). If the computed value looks close to one of the sides, the triangle is likely very \"flat\" near that vertex.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Which angle is bisected:<\/strong> the calculator always bisects the angle between <strong>Side 1<\/strong> and <strong>Side 2<\/strong>; <strong>Side 3<\/strong> must be the side opposite that angle. To bisect a different angle, reorder the inputs accordingly.<\/li>\n<li><strong>Valid triangle required:<\/strong> the three side lengths must satisfy the triangle inequality \u2014 each side must be smaller than the sum of the other two. Check your values first with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/triangle-inequality\/\">Triangle Inequality Checker<\/a>.<\/li>\n<li><strong>Positive lengths only:<\/strong> all inputs must be greater than zero. Zero or negative values do not form a triangle.<\/li>\n<li><strong>Units:<\/strong> the result is expressed in the same units as the inputs. If you enter sides in centimeters, the bisector length is in centimeters.<\/li>\n<li><strong>Internal bisector only:<\/strong> the formula gives the internal bisector. External angle bisectors follow a different formula and are not computed here.<\/li>\n<\/ul>\n<p>If you need to work with the angles themselves rather than side lengths, the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/angle-bisector\/\">Angle Bisector calculator<\/a> handles the related angle computations, and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/2d-vector-angle\/\">2D Vector Angle calculator<\/a> is useful when your triangle is defined by coordinates instead of side lengths.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter the three side lengths of a triangle to find the length of the internal angle bisector \u2014 the segment that splits the angle between Side 1 and Side 2 into two equal parts.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[23],"class_list":["post-61919","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\/61919","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\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61919"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61919\/revisions"}],"predecessor-version":[{"id":66253,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61919\/revisions\/66253"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61919"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61919"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}