{"id":236510,"date":"2025-06-16T09:43:30","date_gmt":"2025-06-16T09:43:30","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=236510"},"modified":"2025-06-16T09:43:32","modified_gmt":"2025-06-16T09:43:32","slug":"how-to-find-circumradius-of-a-triangle-with-help-of-coordinates","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/16\/how-to-find-circumradius-of-a-triangle-with-help-of-coordinates\/","title":{"rendered":"How to find circumradius of a triangle with help of coordinates"},"content":{"rendered":"\n<p>How to find circumradius of a triangle with help of coordinates<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the <strong>circumradius (R)<\/strong> of a triangle using the coordinates of its vertices, use the formula:R=abc4AR = \\frac{abc}{4A}R=4Aabc\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>aaa, bbb, and ccc are the lengths of the sides of the triangle<\/li>\n\n\n\n<li>AAA is the area of the triangle<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-step Method:<\/h3>\n\n\n\n<p>Assume the vertices of the triangle are A(x1,y1)A(x_1, y_1)A(x1\u200b,y1\u200b), B(x2,y2)B(x_2, y_2)B(x2\u200b,y2\u200b), and C(x3,y3)C(x_3, y_3)C(x3\u200b,y3\u200b)<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">1. Calculate the lengths of the sides:<\/h4>\n\n\n\n<p>a=distance&nbsp;between&nbsp;B&nbsp;and&nbsp;C=(x3\u2212x2)2+(y3\u2212y2)2a = \\text{distance between B and C} = \\sqrt{(x_3 &#8211; x_2)^2 + (y_3 &#8211; y_2)^2}a=distance&nbsp;between&nbsp;B&nbsp;and&nbsp;C=(x3\u200b\u2212x2\u200b)2+(y3\u200b\u2212y2\u200b)2\u200bb=distance&nbsp;between&nbsp;A&nbsp;and&nbsp;C=(x3\u2212x1)2+(y3\u2212y1)2b = \\text{distance between A and C} = \\sqrt{(x_3 &#8211; x_1)^2 + (y_3 &#8211; y_1)^2}b=distance&nbsp;between&nbsp;A&nbsp;and&nbsp;C=(x3\u200b\u2212x1\u200b)2+(y3\u200b\u2212y1\u200b)2\u200bc=distance&nbsp;between&nbsp;A&nbsp;and&nbsp;B=(x2\u2212x1)2+(y2\u2212y1)2c = \\text{distance between A and B} = \\sqrt{(x_2 &#8211; x_1)^2 + (y_2 &#8211; y_1)^2}c=distance&nbsp;between&nbsp;A&nbsp;and&nbsp;B=(x2\u200b\u2212x1\u200b)2+(y2\u200b\u2212y1\u200b)2\u200b<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2. Find the area (A) using the determinant method:<\/h4>\n\n\n\n<p>A=12\u2223x1(y2\u2212y3)+x2(y3\u2212y1)+x3(y1\u2212y2)\u2223A = \\frac{1}{2} \\left| x_1(y_2 &#8211; y_3) + x_2(y_3 &#8211; y_1) + x_3(y_1 &#8211; y_2) \\right|A=21\u200b\u2223x1\u200b(y2\u200b\u2212y3\u200b)+x2\u200b(y3\u200b\u2212y1\u200b)+x3\u200b(y1\u200b\u2212y2\u200b)\u2223<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">3. Plug the values into the circumradius formula:<\/h4>\n\n\n\n<p>R=abc4AR = \\frac{abc}{4A}R=4Aabc\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The circumradius is the radius of the circle that passes through all three vertices of a triangle. It does not depend on the triangle being right-angled, scalene, or isosceles. The formula used relates the product of the triangle\u2019s side lengths with its area.<\/p>\n\n\n\n<p>Using coordinates makes it possible to compute side lengths and area algebraically. Side lengths come from the distance formula, and the area comes from a determinant expression derived from the shoelace formula. The final ratio ensures the result gives the radius of the circumscribed circle. This process is especially useful in coordinate geometry and analytic geometry problems involving circles and triangles.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-208.jpeg\" alt=\"\" class=\"wp-image-236511\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>How to find circumradius of a triangle with help of coordinates The Correct Answer and Explanation is: To find the circumradius (R) of a triangle using the coordinates of its vertices, use the formula:R=abc4AR = \\frac{abc}{4A}R=4Aabc\u200b Where: Step-by-step Method: Assume the vertices of the triangle are A(x1,y1)A(x_1, y_1)A(x1\u200b,y1\u200b), B(x2,y2)B(x_2, y_2)B(x2\u200b,y2\u200b), and C(x3,y3)C(x_3, y_3)C(x3\u200b,y3\u200b) 1. Calculate [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-236510","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236510","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=236510"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236510\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=236510"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=236510"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=236510"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}