{"id":67307,"date":"2026-10-21T09:46:54","date_gmt":"2026-10-21T06:46:54","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=67307"},"modified":"2026-10-21T09:46:54","modified_gmt":"2026-10-21T06:46:54","slug":"triangle-calculator-solve-any-triangle","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/triangle-calculator-solve-any-triangle\/","title":{"rendered":"Triangle Calculator \u2013 Solve Any Triangle"},"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\">Solve Any Triangle from Three Known Values<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"67307\" data-favorite-title=\"Triangle\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/triangle-calculator-solve-any-triangle\/\" data-favorite-excerpt=\"Enter any three known values of a triangle \u2014 at least one side \u2014 and instantly get all three sides, all three angles, area, perimeter, heights, inradius, circumradius, plus a step-by-step solution and a scale drawing.\" 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-calculator-solve-any-triangle\/\" method=\"POST\" class=\"calculator\" id=\"calculator-67307\" data-post=\"67307\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Sides<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Side a<\/label><div class=\"term\">a<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"side_a\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"3\"\/><small id=\"calc-info-2\">Fill in exactly three of the six values in total \u2014 at least one must be a side. Same length unit for all sides.<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Side b<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"side_b\" id=\"field-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"4\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Side c<\/label><div class=\"term\">c<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"side_c\" id=\"field-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"5\"\/><\/div><div class=\"is-single field\" id=\"input-5\"><span class=\"font-semibold\">Angles<\/span><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Angle A<\/label><div class=\"term\">A<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"angle_a\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 40\" min=\"0\"\/><small id=\"calc-info-6\">Angle A is opposite side a. SSA data may produce two valid triangles \u2014 both are shown.<\/small><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">Angle B<\/label><div class=\"term\">B<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"angle_b\" id=\"field-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 62\" min=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-8\"><label for=\"field-8\">Angle C<\/label><div class=\"term\">C<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"angle_c\" id=\"field-8\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 78\" min=\"0\"\/><\/div><div class=\"field is-single\" id=\"input-9\"><label for=\"field-9\">Angle unit<\/label><div class=\"grid grid-cols-1\"><select id=\"field-9\" name=\"angle_unit\"><option value=\"deg\" selected>degrees (\u00b0)<\/option><option value=\"rad\">radians (rad)<\/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><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"67307\">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> Solved Triangle<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Solution summary<\/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\" id=\"output-2\"><span class=\"field-title\"><span>Side a<\/span><span class=\"result-term\">a<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"units\"><\/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>Side b<\/span><span class=\"result-term\">b<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"units\"><\/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-4\"><span class=\"field-title\"><span>Side c<\/span><span class=\"result-term\">c<\/span><\/span><span class=\"field-value\" id=\"result-4\" data-unit=\"units\"><\/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-5\"><span class=\"field-title\"><span>Angle A<\/span><span class=\"result-term\">A<\/span><\/span><span class=\"field-value\" id=\"result-5\" 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-6\"><span class=\"field-title\"><span>Angle B<\/span><span class=\"result-term\">B<\/span><\/span><span class=\"field-value\" id=\"result-6\" 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-7\"><span class=\"field-title\"><span>Angle C<\/span><span class=\"result-term\">C<\/span><\/span><span class=\"field-value\" id=\"result-7\" 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-8\"><span class=\"field-title\"><span>Area<\/span><span class=\"result-term\">K<\/span><\/span><span class=\"field-value\" id=\"result-8\" data-unit=\"units\u00b2\"><\/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-9\"><span class=\"field-title\"><span>Perimeter<\/span><span class=\"result-term\">P<\/span><\/span><span class=\"field-value\" id=\"result-9\" data-unit=\"units\"><\/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-10\"><span class=\"field-title\"><span>Height to side a<\/span><span class=\"result-term\">h\u2090<\/span><\/span><span class=\"field-value\" id=\"result-10\" data-unit=\"units\"><\/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-11\"><span class=\"field-title\"><span>Height to side b<\/span><span class=\"result-term\">h_b<\/span><\/span><span class=\"field-value\" id=\"result-11\" data-unit=\"units\"><\/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-12\"><span class=\"field-title\"><span>Height to side c<\/span><span class=\"result-term\">h_c<\/span><\/span><span class=\"field-value\" id=\"result-12\" data-unit=\"units\"><\/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-13\"><span class=\"field-title\"><span>Inradius<\/span><span class=\"result-term\">r<\/span><\/span><span class=\"field-value\" id=\"result-13\" data-unit=\"units\"><\/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-14\"><span class=\"field-title\"><span>Circumradius<\/span><span class=\"result-term\">R<\/span><\/span><span class=\"field-value\" id=\"result-14\" data-unit=\"units\"><\/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-15\"><span class=\"field-title\"><span>Triangle type<\/span><\/span><span class=\"field-value\" id=\"result-15\"><\/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-16\"><span class=\"field-title\"><span>Solution steps<\/span><\/span><span class=\"field-value\" id=\"result-16\"><\/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-17\"><span class=\"field-title\"><span>Triangle to scale<\/span><\/span><span class=\"field-value\" id=\"result-17\"><\/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-18\"><span class=\"field-title\"><span>Second solution (ambiguous case)<\/span><\/span><span class=\"field-value\" id=\"result-18\"><\/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-19\"><span class=\"field-title\"><span>Side a \u2014 2nd triangle<\/span><span class=\"result-term\">a<\/span><\/span><span class=\"field-value\" id=\"result-19\" data-unit=\"units\"><\/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-20\"><span class=\"field-title\"><span>Side b \u2014 2nd triangle<\/span><span class=\"result-term\">b<\/span><\/span><span class=\"field-value\" id=\"result-20\" data-unit=\"units\"><\/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-21\"><span class=\"field-title\"><span>Side c \u2014 2nd triangle<\/span><span class=\"result-term\">c<\/span><\/span><span class=\"field-value\" id=\"result-21\" data-unit=\"units\"><\/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-22\"><span class=\"field-title\"><span>Angle A \u2014 2nd triangle<\/span><span class=\"result-term\">A<\/span><\/span><span class=\"field-value\" id=\"result-22\" 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-23\"><span class=\"field-title\"><span>Angle B \u2014 2nd triangle<\/span><span class=\"result-term\">B<\/span><\/span><span class=\"field-value\" id=\"result-23\" 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-24\"><span class=\"field-title\"><span>Angle C \u2014 2nd triangle<\/span><span class=\"result-term\">C<\/span><\/span><span class=\"field-value\" id=\"result-24\" 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-25\"><span class=\"field-title\"><span>Area \u2014 2nd triangle<\/span><span class=\"result-term\">K<\/span><\/span><span class=\"field-value\" id=\"result-25\" data-unit=\"units\u00b2\"><\/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-26\"><span class=\"field-title\"><span>Perimeter \u2014 2nd triangle<\/span><span class=\"result-term\">P<\/span><\/span><span class=\"field-value\" id=\"result-26\" data-unit=\"units\"><\/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-27\"><span class=\"field-title\"><span>Second triangle to scale<\/span><\/span><span class=\"field-value\" id=\"result-27\"><\/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=\"67307\" 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 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\\[ \\text{Law of cosines (SSS \/ SAS):} \\quad a^2 = b^2 + c^2 - 2bc\\cos A \\]\n\\[ \\cos A = \\frac{b^2 + c^2 - a^2}{2bc}, \\qquad \\cos B = \\frac{a^2 + c^2 - b^2}{2ac}, \\qquad \\cos C = \\frac{a^2 + b^2 - c^2}{2ab} \\\\\n\\text{Law of sines (SSA \/ ASA \/ AAS):} \\quad \\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C} \\\\\n\\text{Angle sum:} \\quad A + B + C = 180^\\circ \\\\\n\\text{Derived measures:} \\quad K = \\tfrac{1}{2}ab\\sin C, \\quad P = a + b + c, \\quad h_a = \\frac{2K}{a}, \\quad r = \\frac{2K}{P}, \\quad R = \\frac{a}{2\\sin A}\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a, b, c\\) \u2014 sides of the triangle; side \\(a\\) is opposite angle \\(A\\), and so on<\/li>\n<li>\\(A, B, C\\) \u2014 interior angles in degrees<\/li>\n<li>\\(K\\) \u2014 area, \\(P\\) \u2014 perimeter, \\(h_a\\) \u2014 height to side \\(a\\)<\/li>\n<li>\\(r\\) \u2014 inradius (radius of the inscribed circle), \\(R\\) \u2014 circumradius (radius of the circumscribed circle)<\/li>\n<\/ul>\n<p>Which formula applies depends on what you know: the law of cosines handles three sides (SSS) or two sides with the included angle (SAS), while the law of sines and the 180\u00b0 angle sum handle the remaining cases.<\/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 \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> the default values \u2014 sides \\(a = 3\\), \\(b = 4\\) and \\(c = 5\\), with all angle fields left empty. Clear the angle fields, keep the three sides, and the calculator solves this triangle as an SSS case.<\/p>\n<p>\\[ \\cos C = \\frac{a^2 + b^2 - c^2}{2ab} = \\frac{3^2 + 4^2 - 5^2}{2 \\cdot 3 \\cdot 4} = \\frac{9 + 16 - 25}{24} = 0 \\quad \\Rightarrow \\quad C = 90^\\circ \\]<\/p>\n<p>\\[ \\cos A = \\frac{b^2 + c^2 - a^2}{2bc} = \\frac{16 + 25 - 9}{40} = 0.8 \\quad \\Rightarrow \\quad A \\approx 36.87^\\circ \\]<\/p>\n<p>\\[ B = 180^\\circ - 90^\\circ - 36.87^\\circ = 53.13^\\circ \\]<\/p>\n<p>\\[ K = \\tfrac{1}{2}ab\\sin C = \\tfrac{1}{2} \\cdot 3 \\cdot 4 \\cdot \\sin 90^\\circ = 6, \\quad P = 3 + 4 + 5 = 12 \\]<\/p>\n<p>\\[ r = \\frac{2K}{P} = \\frac{12}{12} = 1, \\qquad R = \\frac{a}{2\\sin A} = \\frac{3}{2 \\cdot 0.6} = 2.5 \\]<\/p>\n<p><strong>Result:<\/strong> an angled, right scalene triangle with \\(A \\approx 36.87^\\circ\\), \\(B \\approx 53.13^\\circ\\), \\(C = 90^\\circ\\), area 6 units\u00b2, perimeter 12 units, heights \\(h_a = 4\\), \\(h_b = 3\\), \\(h_c = 2.4\\), inradius 1 and circumradius 2.5. This is the famous 3-4-5 right triangle.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Fill in <strong>exactly three<\/strong> of the six fields: <strong>Side a<\/strong>, <strong>Side b<\/strong>, <strong>Side c<\/strong>, <strong>Angle A<\/strong>, <strong>Angle B<\/strong> or <strong>Angle C<\/strong>. Leave the other three empty.<\/li>\n<li>Make sure at least one of the three entries is a side \u2014 three angles alone fix only the shape of a triangle, not its size.<\/li>\n<li>Use the same length unit for all sides; the results inherit that unit.<\/li>\n<li>If your angles are in radians, set <strong>Angle unit<\/strong> to <em>radians (rad)<\/em>. All results are always displayed in degrees.<\/li>\n<li>Check the <em>Solved Triangle<\/em> block: it lists all six measures, the area, the perimeter, the three heights, both radii, the triangle type and a scale drawing, followed by a step-by-step solution.<\/li>\n<li>If you entered two sides and a non-included angle (the SSA case), the calculator shows both valid triangles when they exist \u2014 compare the two drawings and pick the one that matches your problem.<\/li>\n<\/ol>\n<h2>What does it mean to \"solve\" a triangle?<\/h2>\n<p>A triangle has six main parts: three sides and three angles. Solving a triangle means finding all six from the three you already know. Not any three values work, however \u2014 three angles determine only a family of similar triangles, which is why the calculator requires at least one side. Three sides only work when they satisfy the triangle inequality: each side must be shorter than the sum of the other two.<\/p>\n<p>Every valid combination falls into one of four classical cases, and each has a standard route:<\/p>\n<ul>\n<li><strong>SSS<\/strong> (three sides): each angle follows from the law of cosines.<\/li>\n<li><strong>SAS<\/strong> (two sides and the angle <em>between<\/em> them): the third side comes from the law of cosines, then the angles from the law of cosines or the 180\u00b0 sum.<\/li>\n<li><strong>ASA \/ AAS<\/strong> (one side and two angles): the third angle comes from the 180\u00b0 sum, then all sides from the law of sines.<\/li>\n<li><strong>SSA<\/strong> (two sides and an angle <em>not<\/em> between them): solved with the law of sines, but this case is special \u2014 see below.<\/li>\n<\/ul>\n<p>The result output goes beyond the six basic measures. The area uses \\(K = \\tfrac{1}{2}ab\\sin C\\), each height is \\(h = 2K \/ \\text{side}\\), the inradius \\(r\\) is the radius of the circle that touches all three sides, and the circumradius \\(R\\) is the radius of the circle passing through all three vertices. Knowing \\(r\\) is useful for packing and clearance problems; \\(R\\) appears constantly in trigonometry and geometry proofs. If you only need the area of a triangle, the dedicated <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/triangle-area-calculator\/\">triangle area calculator<\/a> is a quicker tool.<\/p>\n<h2>The ambiguous case (SSA): two triangles from the same data<\/h2>\n<p>When you know two sides and an angle <em>opposite<\/em> one of them, the data may not determine a unique triangle. Imagine a door of fixed length swinging on a hinge: for a certain range of positions, its free end can touch a given line at two different points, producing two different triangles with exactly the same \\(a\\), \\(b\\) and \\(A\\).<\/p>\n<p>Concretely, if the known angle is acute and the side opposite it is shorter than the other known side (but not shorter than the altitude), the law of sines gives two possible values for the remaining angle: \\(B\\) and \\(180^\\circ - B\\). The calculator detects this automatically and outputs a complete second solution \u2014 sides, angles, area, perimeter and its own drawing \u2014 so you can decide which triangle your problem refers to. In all other situations (the angle is 90\u00b0 or larger, or the opposite side is the longest), the triangle is unique.<\/p>\n<h2>Assumptions, limits and practical tips<\/h2>\n<ul>\n<li><strong>Plain (Euclidean) triangles only.<\/strong> Angles must be greater than 0\u00b0 and smaller than 180\u00b0, and side lengths must be positive.<\/li>\n<li><strong>Exactly three values.<\/strong> Entering more than three makes the system overdetermined \u2014 if you know four or more parts, cross-check them by solving with different triples; a genuine triangle gives consistent results each time.<\/li>\n<li><strong>Angle matching matters.<\/strong> Angle \\(A\\) must sit opposite side \\(a\\), angle \\(B\\) opposite side \\(b\\), and so on. Mixing them up is the most common source of a \"wrong\" answer.<\/li>\n<li><strong>Units are up to you.<\/strong> Sides can be in cm, inches, meters \u2014 anything \u2014 provided all three share one unit; the area is then in that unit squared.<\/li>\n<li><strong>No tolerance for near-degenerate triangles.<\/strong> If the sides almost violate the triangle inequality, or the angles almost sum to 180\u00b0, the tool reports an error rather than a nearly flat triangle.<\/li>\n<\/ul>\n<p>For a right triangle where you only work with the two legs, the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/right-triangle-calculator\/\">right triangle calculator<\/a> is simpler; the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/pythagorean-theorem-calculator\/\">Pythagorean theorem calculator<\/a> covers just the side relationships, and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/trigonometry-calculator\/\">trigonometry calculator<\/a> evaluates the individual sine, cosine and tangent values behind these formulas.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter any three known values of a triangle \u2014 at least one side \u2014 and instantly get all three sides, all three angles, area, perimeter, heights, inradius, circumradius, plus a step-by-step solution and a scale drawing.<\/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-67307","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\/67307","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=67307"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67307\/revisions"}],"predecessor-version":[{"id":67308,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67307\/revisions\/67308"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=67307"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=67307"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=67307"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}