{"id":249102,"date":"2025-07-08T22:32:28","date_gmt":"2025-07-08T22:32:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=249102"},"modified":"2025-07-08T22:32:30","modified_gmt":"2025-07-08T22:32:30","slug":"calculate-the-centroid-of-a-semicircle","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/08\/calculate-the-centroid-of-a-semicircle\/","title":{"rendered":"Calculate the centroid of a semicircle"},"content":{"rendered":"\n

Calculate the centroid of a semicircle.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The centroid (or center of mass) of a semicircle can be determined using integral calculus. For a semicircle, the centroid lies along the central axis of symmetry, which is the vertical line passing through the center of the full circle. For a semicircle with radius RRR, we will calculate the distance of the centroid from the flat edge of the semicircle, which lies along the x-axis.<\/p>\n\n\n\n

Step-by-step Explanation:<\/h3>\n\n\n\n
    \n
  1. Equation of the Semicircle<\/strong>:
    A semicircle with radius RRR can be represented by the equation: y=R2\u2212x2y = \\sqrt{R^2 – x^2}y=R2\u2212x2\u200b where \u2212R\u2264x\u2264R-R \\leq x \\leq R\u2212R\u2264x\u2264R, and the semicircle is above the x-axis.<\/li>\n\n\n\n
  2. Finding the Centroid<\/strong>:
    The centroid of a region is given by the formula: y\u02c9=\u222b\u2212RRy\u2009dA\u222b\u2212RRdA\\bar{y} = \\frac{\\int_{-R}^{R} y \\, dA}{\\int_{-R}^{R} dA}y\u02c9\u200b=\u222b\u2212RR\u200bdA\u222b\u2212RR\u200bydA\u200b where dAdAdA is the differential area element. For the semicircle, dA=y\u2009dxdA = y \\, dxdA=ydx, so we substitute this into the formula.<\/li>\n\n\n\n
  3. Calculating the Area<\/strong>:
    The total area of the semicircle is: A=\u222b\u2212RRy\u2009dx=\u222b\u2212RRR2\u2212x2\u2009dxA = \\int_{-R}^{R} y \\, dx = \\int_{-R}^{R} \\sqrt{R^2 – x^2} \\, dxA=\u222b\u2212RR\u200bydx=\u222b\u2212RR\u200bR2\u2212x2\u200bdx This is a standard integral, and its result is: A=12\u03c0R2A = \\frac{1}{2} \\pi R^2A=21\u200b\u03c0R2<\/li>\n\n\n\n
  4. Finding the Centroid’s yyy-coordinate<\/strong>:
    Now, we calculate the centroid’s yyy-coordinate: y\u02c9=\u222b\u2212RRy2\u2009dx\u222b\u2212RRy\u2009dx\\bar{y} = \\frac{\\int_{-R}^{R} y^2 \\, dx}{\\int_{-R}^{R} y \\, dx}y\u02c9\u200b=\u222b\u2212RR\u200bydx\u222b\u2212RR\u200by2dx\u200b First, calculate the integral of y2y^2y2: \u222b\u2212RRy2\u2009dx=\u222b\u2212RR(R2\u2212x2)\u2009dx=\u03c0R48\\int_{-R}^{R} y^2 \\, dx = \\int_{-R}^{R} (R^2 – x^2) \\, dx = \\frac{\\pi R^4}{8}\u222b\u2212RR\u200by2dx=\u222b\u2212RR\u200b(R2\u2212x2)dx=8\u03c0R4\u200b Now, divide the integral of y2y^2y2 by the area of the semicircle: y\u02c9=\u03c0R4812\u03c0R2=4R3\u03c0\\bar{y} = \\frac{\\frac{\\pi R^4}{8}}{\\frac{1}{2} \\pi R^2} = \\frac{4R}{3\\pi}y\u02c9\u200b=21\u200b\u03c0R28\u03c0R4\u200b\u200b=3\u03c04R\u200b<\/li>\n<\/ol>\n\n\n\n

    Final Result:<\/h3>\n\n\n\n

    The centroid of a semicircle is located at a distance of 4R3\u03c0\\frac{4R}{3\\pi}3\u03c04R\u200b from the flat edge along the vertical axis of symmetry. This is approximately 0.424 of the radius from the flat edge.<\/p>\n\n\n\n

    Thus, the centroid lies 4R3\u03c0\\frac{4R}{3\\pi}3\u03c04R\u200b units above the flat edge of the semicircle<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Calculate the centroid of a semicircle. The Correct Answer and Explanation is: The centroid (or center of mass) of a semicircle can be determined using integral calculus. For a semicircle, the centroid lies along the central axis of symmetry, which is the vertical line passing through the center of the full circle. For a semicircle […]<\/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-249102","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/249102","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=249102"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/249102\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=249102"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=249102"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=249102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}