{"id":36850,"date":"2025-06-25T09:50:36","date_gmt":"2025-06-25T09:50:36","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=36850"},"modified":"2025-06-25T09:50:37","modified_gmt":"2025-06-25T09:50:37","slug":"find-the-exact-area-of-the-shaded-region","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-area-of-the-shaded-region\/","title":{"rendered":"Find the exact area of the shaded region"},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"640\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-319.png\" alt=\"\" class=\"wp-image-36852\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-319.png 640w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-319-300x300.png 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-319-150x150.png 150w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The correct answer is <strong>(192\u03c0 \u2212 144\u221a3) m\u00b2<\/strong>.<\/p>\n\n\n\n<p>To find the exact area of the shaded region, we begin with the area of the sector and subtract the area of the triangle formed by the two radii and the chord connecting their endpoints. This shaded region resembles a segment of the circle, not the full sector.<\/p>\n\n\n\n<p>Start by calculating the area of the sector. The formula for the area of a sector is:<\/p>\n\n\n\n<p>Asector=\u03b8360\u22c5\u03c0r2A_{\\text{sector}} = \\frac{\u03b8}{360} \\cdot \u03c0r\u00b2<\/p>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Radius r=24r = 24 meters<\/li>\n\n\n\n<li>Central angle \u03b8=120\u2218\u03b8 = 120^\\circ<\/li>\n<\/ul>\n\n\n\n<p>Asector=120360\u22c5\u03c0\u22c5242=13\u22c5\u03c0\u22c5576=192\u03c0&nbsp;m\u00b2A_{\\text{sector}} = \\frac{120}{360} \\cdot \u03c0 \\cdot 24\u00b2 = \\frac{1}{3} \\cdot \u03c0 \\cdot 576 = 192\u03c0 \\text{ m\u00b2}<\/p>\n\n\n\n<p>Next, calculate the area of the triangle formed by the two radii and the included 120\u00b0 angle. This triangle is not a right triangle, so we use the sine area formula:<\/p>\n\n\n\n<p>Atriangle=12absin\u2061(C)A_{\\text{triangle}} = \\frac{1}{2} ab \\sin(C)<\/p>\n\n\n\n<p>Here, a=b=24a = b = 24, and angle C=120\u2218C = 120^\\circ:<\/p>\n\n\n\n<p>Atriangle=12\u22c524\u22c524\u22c5sin\u2061(120\u2218)=12\u22c5576\u22c532=1443&nbsp;m\u00b2A_{\\text{triangle}} = \\frac{1}{2} \\cdot 24 \\cdot 24 \\cdot \\sin(120^\\circ) = \\frac{1}{2} \\cdot 576 \\cdot \\frac{\\sqrt{3}}{2} = 144\\sqrt{3} \\text{ m\u00b2}<\/p>\n\n\n\n<p>Now subtract the area of the triangle from the sector:<\/p>\n\n\n\n<p>Ashaded=192\u03c0\u22121443&nbsp;m\u00b2A_{\\text{shaded}} = 192\u03c0 \u2212 144\\sqrt{3} \\text{ m\u00b2}<\/p>\n\n\n\n<p>This result represents the area of the circular segment bounded by the arc and the chord. It is the exact area of the shaded region shown in the image.<\/p>\n\n\n\n<p>Hence, the correct expression is: <strong>(192\u03c0 \u2212 144\u221a3) m\u00b2<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-819.jpeg\" alt=\"\" class=\"wp-image-36856\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-819.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-819-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-819-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The Correct Answer and Explanation is: The correct answer is (192\u03c0 \u2212 144\u221a3) m\u00b2. To find the exact area of the shaded region, we begin with the area of the sector and subtract the area of the triangle formed by the two radii and the chord connecting their endpoints. This shaded region resembles a segment [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-36850","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36850","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=36850"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36850\/revisions"}],"predecessor-version":[{"id":36857,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36850\/revisions\/36857"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=36850"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=36850"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=36850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}