{"id":61460,"date":"2021-02-02T16:03:13","date_gmt":"2021-02-02T14:03:13","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61460"},"modified":"2026-09-30T20:51:21","modified_gmt":"2026-09-30T17:51:21","slug":"triangle-inequality","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/triangle-inequality\/","title":{"rendered":"Triangle Inequality Checker"},"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\">Check the Triangle Inequality<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61460\" data-favorite-title=\"Triangle Inequality Checker\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/triangle-inequality\/\" data-favorite-excerpt=\"Check whether three side lengths can form a valid triangle. Enter the values for sides a, b, and c, and the calculator applies the Triangle Inequality Theorem to tell you if a triangle is possible.\" 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\/triangle-inequality\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61460\" data-post=\"61460\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Side a<\/label><input type=\"text\" name=\"side_a\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"1\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Side b<\/label><input type=\"text\" name=\"side_b\" id=\"field-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"6\" min=\"1\" value=\"6\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Side c<\/label><input type=\"text\" name=\"side_c\" id=\"field-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"10\" min=\"1\" value=\"10\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61460\">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>Triangle Inequality Result<\/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=\"61460\" 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>Triangle Inequality Theorem 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\\[a + b &gt; c\\] \\[a + c &gt; b\\] \\[b + c &gt; a\\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a\\), \\(b\\), \\(c\\) \u2014 the lengths of the three sides<\/li>\n<\/ul>\n<p>According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. All three inequalities must hold; if even one fails, the sides cannot form a triangle.<\/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>Triangle Inequality \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> check whether the sides \\(a = 5\\), \\(b = 6\\), and \\(c = 10\\) can form a triangle.<\/p>\n<p>\\[a + b = 5 + 6 = 11 &gt; 10 \\;\\checkmark\\]<\/p>\n<p>\\[a + c = 5 + 10 = 15 &gt; 6 \\;\\checkmark\\]<\/p>\n<p>\\[b + c = 6 + 10 = 16 &gt; 5 \\;\\checkmark\\]<\/p>\n<p><strong>Result:<\/strong> all three inequalities are satisfied, so a triangle with sides 5, 6, and 10 is possible.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the length of <strong>Side a<\/strong> in the first field.<\/li>\n<li>Enter the length of <strong>Side b<\/strong> in the second field.<\/li>\n<li>Enter the length of <strong>Side c<\/strong> in the third field.<\/li>\n<li>Read the <strong>Triangle Inequality Result<\/strong>: the calculator tells you whether a triangle with those three sides is possible.<\/li>\n<\/ol>\n<p>All side lengths must be positive numbers. If any of the three inequalities fails, the result will state that a triangle is not possible.<\/p>\n<h2>What Is the Triangle Inequality Theorem?<\/h2>\n<p>The Triangle Inequality Theorem is one of the fundamental rules of Euclidean geometry. It states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Intuitively, this reflects the fact that a straight line is the shortest path between two points: if one side were as long as or longer than the sum of the other two, the two shorter sides would have to lie flat along the third side, and the figure would collapse into a straight line segment rather than a triangle.<\/p>\n<p>It is enough to verify the single largest side against the sum of the two smaller ones. If the largest side is shorter than the sum of the other two, the remaining two inequalities hold automatically. The calculator checks all three conditions, which is a robust way to test sides regardless of which one is largest.<\/p>\n<p>The theorem has practical uses beyond geometry class. Engineers and architects use it when verifying that structural elements can close into a triangular frame; surveyors rely on it when checking that measured distances are consistent; and programmers use it in computer graphics and physics engines to validate geometric data. It also underpins the definition of distance in mathematics: any valid distance function must satisfy a triangle inequality of its own.<\/p>\n<p>Note that the theorem only checks <em>feasibility<\/em>. It tells you that a triangle exists, but not its angles or area. If you need those, the sides determine the triangle completely, so you can go on to compute its angles with the Law of Cosines or construct auxiliary lines such as an <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/angle-bisector\/\">angle bisector<\/a> or the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/length-angle-bisector-calculator\/\">length of an angle bisector<\/a>.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Strict inequality:<\/strong> if the sum of two sides equals the third (for example, 3, 4, and 7), the result is a degenerate \"triangle\" \u2014 a straight line. The theorem requires a strict <em>greater than<\/em>, so such input is reported as impossible.<\/li>\n<li><strong>Positive lengths only:<\/strong> side lengths must be positive real numbers. Zero or negative values have no geometric meaning for a triangle.<\/li>\n<li><strong>Equal sides are valid:<\/strong> an equilateral triangle (e.g., 5, 5, 5) or an isosceles triangle (e.g., 5, 5, 8) passes the test as long as the inequalities hold.<\/li>\n<li><strong>Units:<\/strong> the calculator works with any unit of length (cm, inches, meters), but all three sides must use the same unit for the comparison to be meaningful.<\/li>\n<li><strong>It does not compute angles or area:<\/strong> if you also need the triangle's dimensions or related vectors, see tools like the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/2d-vector-magnitude\/\">2D Vector Magnitude calculator<\/a> or the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/2d-vector-angle\/\">2D Vector Angle calculator<\/a> for follow-up computations.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Check whether three side lengths can form a valid triangle. Enter the values for sides a, b, and c, and the calculator applies the Triangle Inequality Theorem to tell you if a triangle is possible.<\/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-61460","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\/61460","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=61460"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61460\/revisions"}],"predecessor-version":[{"id":66297,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61460\/revisions\/66297"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61460"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61460"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}