{"id":233753,"date":"2025-06-13T11:24:33","date_gmt":"2025-06-13T11:24:33","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233753"},"modified":"2025-06-13T11:24:36","modified_gmt":"2025-06-13T11:24:36","slug":"calc-area-of-segment-fractheta360-pi-r2-frac12r2-times-fracthetafracpi2-frac2111","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/calc-area-of-segment-fractheta360-pi-r2-frac12r2-times-fracthetafracpi2-frac2111\/","title":{"rendered":"Calc: Area of Segment = \\frac{\\theta}{360} \\pi r^2 – \\frac{1}{2}r^2 \\times \\frac{\\theta}{\\frac{\\pi}{2}} \\frac{21}{11} ="},"content":{"rendered":"\n

Calc: Area of Segment = \\frac{\\theta}{360} \\pi r^2 – \\frac{1}{2}r^2 \\times \\frac{\\theta}{\\frac{\\pi}{2}} \\frac{21}{11} =<\/p>\n\n\n\n

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

To solve the expression for the area of a segment<\/strong>, we need to carefully understand and simplify the given mathematical formula:Area of Segment=\u03b8360\u03c0r2\u221212r2\u00d7\u03b8\u03c02\u00d72111\\text{Area of Segment} = \\frac{\\theta}{360} \\pi r^2 – \\frac{1}{2} r^2 \\times \\frac{\\theta}{\\frac{\\pi}{2}} \\times \\frac{21}{11}Area of Segment=360\u03b8\u200b\u03c0r2\u221221\u200br2\u00d72\u03c0\u200b\u03b8\u200b\u00d71121\u200b<\/p>\n\n\n\n

\n\n\n\n

Step 1: Understand the formula<\/h3>\n\n\n\n

The area of a segment of a circle<\/strong> is given by:Area of Segment=Area of Sector\u2212Area of Triangle\\text{Area of Segment} = \\text{Area of Sector} – \\text{Area of Triangle}Area of Segment=Area of Sector\u2212Area of Triangle<\/p>\n\n\n\n

    \n
  • Area of Sector<\/strong>: \u03b8360\u03c0r2\\frac{\\theta}{360} \\pi r^2360\u03b8\u200b\u03c0r2, where \u03b8\\theta\u03b8 is in degrees.<\/li>\n\n\n\n
  • Area of Triangle<\/strong>: often found using trigonometry, or other geometric approximations.<\/li>\n<\/ul>\n\n\n\n

    However, the second part of your expression is unusual:12r2\u00d7\u03b8\u03c02\u00d72111\\frac{1}{2} r^2 \\times \\frac{\\theta}{\\frac{\\pi}{2}} \\times \\frac{21}{11}21\u200br2\u00d72\u03c0\u200b\u03b8\u200b\u00d71121\u200b<\/p>\n\n\n\n

    Let\u2019s first simplify this expression step-by-step assuming \u03b8=60\u2218\\theta = 60^\\circ\u03b8=60\u2218 and r=7r = 7r=7 (example values \u2014 if you have actual values, please provide them).<\/p>\n\n\n\n

    \n\n\n\n

    Step 2: Substitute values (assume r=7r = 7r=7, \u03b8=60\\theta = 60\u03b8=60)<\/h3>\n\n\n\n

    First Term<\/strong>:60360\u03c0(7)2=16\u03c0\u00d749=49\u03c06\\frac{60}{360} \\pi (7)^2 = \\frac{1}{6} \\pi \\times 49 = \\frac{49\\pi}{6}36060\u200b\u03c0(7)2=61\u200b\u03c0\u00d749=649\u03c0\u200b<\/p>\n\n\n\n

    Second Term<\/strong>:12\u00d749\u00d7(60\u03c02)\u00d72111\\frac{1}{2} \\times 49 \\times \\left( \\frac{60}{\\frac{\\pi}{2}} \\right) \\times \\frac{21}{11}21\u200b\u00d749\u00d7(2\u03c0\u200b60\u200b)\u00d71121\u200b<\/p>\n\n\n\n

    Simplify inside:60\u03c02=60\u00d72\u03c0=120\u03c0\\frac{60}{\\frac{\\pi}{2}} = \\frac{60 \\times 2}{\\pi} = \\frac{120}{\\pi}2\u03c0\u200b60\u200b=\u03c060\u00d72\u200b=\u03c0120\u200b<\/p>\n\n\n\n

    So the second term becomes:12\u00d749\u00d7120\u03c0\u00d72111=49\u00d7120\u00d7212\u00d711\u03c0=12348022\u03c0=5612.73\u03c0\u22481786.87\\frac{1}{2} \\times 49 \\times \\frac{120}{\\pi} \\times \\frac{21}{11} = \\frac{49 \\times 120 \\times 21}{2 \\times 11 \\pi} = \\frac{123480}{22\\pi} = \\frac{5612.73}{\\pi} \\approx 1786.8721\u200b\u00d749\u00d7\u03c0120\u200b\u00d71121\u200b=2\u00d711\u03c049\u00d7120\u00d721\u200b=22\u03c0123480\u200b=\u03c05612.73\u200b\u22481786.87<\/p>\n\n\n\n

    First Term<\/strong> (numeric):49\u03c06\u2248153.9386\u224825.656\\frac{49\\pi}{6} \\approx \\frac{153.938}{6} \\approx 25.656649\u03c0\u200b\u22486153.938\u200b\u224825.656<\/p>\n\n\n\n

    \n\n\n\n

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

    Area of Segment\u224825.656\u22121786.87=\u22121761.21\\text{Area of Segment} \\approx 25.656 – 1786.87 = -1761.21Area of Segment\u224825.656\u22121786.87=\u22121761.21<\/p>\n\n\n\n

    \n\n\n\n

    Explanation (like in textbooks):<\/h3>\n\n\n\n

    To calculate the area of a segment of a circle<\/strong>, we subtract the area of the triangle formed by the chord from the area of the sector defined by the central angle.<\/p>\n\n\n\n

      \n
    • The area of the sector<\/strong> depends on the angle \u03b8\\theta\u03b8 and the radius rrr of the circle.<\/li>\n\n\n\n
    • The area of the triangle<\/strong> is often found using trigonometric identities or geometric relationships. In this case, a custom expression involving \u03b8\u03c02\u00d72111\\frac{\\theta}{\\frac{\\pi}{2}} \\times \\frac{21}{11}2\u03c0\u200b\u03b8\u200b\u00d71121\u200b is used, suggesting a unit conversion or proportion adjustment.<\/li>\n<\/ul>\n\n\n\n

      In general, we ensure that angle units are consistent (radians or degrees), and then apply formulas carefully, simplifying each component step by step. The negative result here implies a misinterpretation or incorrect application, as area cannot be negative. Likely, the constants or the form of the second term need to be reviewed or clarified.<\/p>\n\n\n\n

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

      Calc: Area of Segment = \\frac{\\theta}{360} \\pi r^2 – \\frac{1}{2}r^2 \\times \\frac{\\theta}{\\frac{\\pi}{2}} \\frac{21}{11} = The Correct Answer and Explanation is: To solve the expression for the area of a segment, we need to carefully understand and simplify the given mathematical formula:Area of Segment=\u03b8360\u03c0r2\u221212r2\u00d7\u03b8\u03c02\u00d72111\\text{Area of Segment} = \\frac{\\theta}{360} \\pi r^2 – \\frac{1}{2} r^2 \\times \\frac{\\theta}{\\frac{\\pi}{2}} \\times \\frac{21}{11}Area of Segment=360\u03b8\u200b\u03c0r2\u221221\u200br2\u00d72\u03c0\u200b\u03b8\u200b\u00d71121\u200b Step […]<\/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-233753","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233753","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=233753"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233753\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233753"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233753"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233753"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}