{"id":252583,"date":"2025-04-09T13:12:37","date_gmt":"2025-04-09T18:12:37","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=252583"},"modified":"2026-03-30T15:42:29","modified_gmt":"2026-03-30T20:42:29","slug":"law-of-sines-calculator","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/law-of-sines-calculator\/","title":{"rendered":"Law of Sines Calculator"},"content":{"rendered":"<p>Use this calculator to find one missing side or angle in a triangle.<\/p>\n<div id=\"sinesCalculator\" class=\"calculatorcontainer\">\n<div class=\"background\">\n<div id=\"leftAndRight\">\n<div id=\"selectAndInputs\">\n<div class=\"labelInputCombo\"><label for=\"dropdown\">Solve for:<\/label><select id=\"dropdown\" onchange=\"changeFields()\"><option value=\"alpha\">Angle \u03b1<\/option><option value=\"sideA\">Side a<\/option><option value=\"sideB\">Side b<\/option><option value=\"beta\">Angle \u03b2<\/option><\/select><\/div>\n<div id=\"inputFields\">\n<div class=\"labelInputCombo\"><label for=\"sideAInput\">Side $a$<\/label><input id=\"sideAInput\" type=\"number\" placeholder=\"Length\" \/><\/div>\n<div class=\"labelInputCombo\"><label for=\"sideBInput\">Side $b$<\/label><input id=\"sideBInput\" type=\"number\" placeholder=\"Length\" \/><\/div>\n<div class=\"labelInputCombo\"><label for=\"betaInput\">Angle \u03b2<\/label><input id=\"betaInput\" type=\"number\" placeholder=\"Degrees\" \/><\/div>\n<\/div>\n<\/div>\n<div id=\"imageAndResult\">\n<div class=\"result\"><\/div>\n<div id=\"image\"><img decoding=\"async\" id=\"triangleImage\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Blank.svg\" alt=\"a triangle with two sides labeled a and b, with angles alpha and beta opposite each respectively.\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"info\"><\/div>\n<\/div>\n<div id=\"bugLink\" style=\"text-align: center; margin-bottom: 20px; font-size: 80%;\">Found a bug? <a class=\"ylist\" href=\"https:\/\/airtable.com\/appgcc1PP0BbzPCbI\/shrb33jgGwb66DWHC?prefill_Test=Law%20Of%20Sines%20Calculator&#038;prefill_Question+Number=0&#038;prefill_Which+Form=Calculator-Feedback&#038;prefill_UP-ID=Calculator-Feedback%20-%20Law%20Of%20Sines%20Calculator%20-%200&#038;hide_Test=true&#038;hide_Which+Form=true&#038;hide_UP-ID=true&#038;hide_Question+Number=true\" target=\"_blank\" rel=\"noopener\">Let us know!<\/a><\/div>\n<\/div>\n<p>Knowing how to use the law of sines is a fundamental skill in trigonometry! Here&#8217;s a quick review:<\/p>\n<h3>What is the Law of Sines?<\/h3>\n<p>The law of sines established a relationship between the sides and angles of any triangle. It states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides.<\/p>\n<p>This is the law of sines formula:<\/p>\n<p style=\"text-align: center;\">$$\\dfrac{a}{\\sin(A)}=\\dfrac{b}{\\sin(B)}=\\dfrac{c}{\\sin(C)}$$<\/p>\n<p>The lowercase letters represent the side lengths, and the uppercase letters represent the angles that are opposite those sides.<\/p>\n<h3>Example: Finding a Missing Side<\/h3>\n<p>Let&#8217;s find the missing length of side $b$ in a triangle where we know the following:<\/p>\n<div class=\"sines-example-container\">\n<div class=\"sines-example-text\">\n<ul>\n<li style=\"margin-bottom: 12px;\">$a=10$<\/li>\n<li style=\"margin-bottom: 12px;\">$\\alpha=40\u00b0$<\/li>\n<li>$\\beta=60\u00b0$<\/li>\n<\/ul>\n<\/div>\n<div class=\"sines-example-image\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/08\/Side-B-unknown.svg\" alt=\"Triangle where side b is unknown\" \/><\/div>\n<\/div>\n<p>We only need to use parts of the formula that involve our known values and the side we want to find, which looks like this:<\/p>\n<p style=\"text-align: center;\">$$\\dfrac{a}{\\sin(\\alpha)}=\\dfrac{b}{\\sin(\\beta)}$$<\/p>\n<p>To get $b$ by itself, we need to multiply both sides of the equation by $\\sin(\\beta)$:<\/p>\n<p style=\"text-align: center;\">$$b=\\dfrac{a\\times \\sin(\\beta)}{\\sin(\\alpha)}$$<\/p>\n<p>Next, substitute the given measurements into the rearranged formula:<\/p>\n<p style=\"text-align: center;\">$$b=\\dfrac{10\\times \\sin(60\u00b0)}{\\sin(40\u00b0)}$$<\/p>\n<p>Finally, find the sine values and solve the equation:<\/p>\n<p style=\"text-align: center;\">$$b \\approx \\dfrac{10\\times 0.866}{0.643}\\approx 13.47$$<\/p>\n<p>Therefore, the length of Side $b$ is around 13.47.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-265990\" role=\"img\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/08\/Side-B-solved.svg\" alt=\"Triangle where side b is approximately 13.47\" width=\"292.1\" height=\"177.1\" \/><\/p>\n<div class=\"home-buttons2\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/law-of-sines\/\">Learn More About the Law of Sines Here!<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Use this calculator to find one missing side or angle in a triangle. Solve for:Angle \u03b1Side aSide bAngle \u03b2 Side $a$ Side $b$ Angle \u03b2 Found a bug? Let us know! Knowing how to use the law of sines is a fundamental skill in trigonometry! Here&#8217;s a quick review: What is the Law of Sines? &#8230; <a title=\"Law of Sines Calculator\" class=\"read-more\" href=\"https:\/\/www.mometrix.com\/academy\/law-of-sines-calculator\/\" aria-label=\"Read more about Law of Sines Calculator\">Read more<\/a><\/p>\n","protected":false},"author":79,"featured_media":266017,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-252583","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_type-calculator"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/252583","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/79"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=252583"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/252583\/revisions"}],"predecessor-version":[{"id":291152,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/252583\/revisions\/291152"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/266017"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=252583"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}