{"id":90222,"date":"2025-06-01T10:23:23","date_gmt":"2025-06-01T10:23:23","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90222"},"modified":"2025-06-01T10:23:23","modified_gmt":"2025-06-01T10:23:23","slug":"an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/","title":{"rendered":"An equilateral triangle is inscribed in a circle of radius 1 unit. The"},"content":{"rendered":"

An equilateral triangle is inscribed in a circle of radius 1 unit. The area of the shaded region, in square unit, is:<\/p>\n

[amp_mcq option1=”\u03c0\/3 – \u221a3\/4″ option2=”\u03c0\/3 – \u221a3\/2″ option3=”\u03c0 – 3″ option4=”\u03c0 – 3\/4″ correct=”option1″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2018<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nAn equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360\u00b0\/3 = 120\u00b0. The area of a sector of the circle corresponding to this angle is (120\u00b0\/360\u00b0) * \u03c0 * r\u00b2 = (1\/3) * \u03c0 * 1\u00b2 = \u03c0\/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1\/2) * r * r * sin(120\u00b0) = (1\/2) * 1 * 1 * (\u221a3\/2) = \u221a3\/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: \u03c0\/3 – \u221a3\/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (\u03c0\/3 – \u221a3\/4) = \u03c0 – 3\u221a3\/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment.
\n<\/section>\n
\nFor an equilateral triangle inscribed in a circle, each side subtends a central angle of 120\u00b0. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.
\n<\/section>\n
\nThe side length of an equilateral triangle inscribed in a circle of radius R is R\u221a3. The area of an equilateral triangle with side ‘a’ is (\u221a3\/4)a\u00b2. For R=1, side a=\u221a3, Area of triangle = (\u221a3\/4)(\u221a3)\u00b2 = 3\u221a3\/4. Area of circle = \u03c0R\u00b2 = \u03c0. Area outside triangle = \u03c0 – 3\u221a3\/4. The question is solvable only by assuming the shaded region is one segment, which is a common type of shaded area in such diagrams.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

An equilateral triangle is inscribed in a circle of radius 1 unit. The area of the shaded region, in square unit, is: [amp_mcq option1=”\u03c0\/3 – \u221a3\/4″ option2=”\u03c0\/3 – \u221a3\/2″ option3=”\u03c0 – 3″ option4=”\u03c0 – 3\/4″ correct=”option1″] This question was previously asked in UPSC CAPF – 2018 Download PDFAttempt Online An equilateral triangle inscribed in a … <\/p>\n

Detailed SolutionAn equilateral triangle is inscribed in a circle of radius 1 unit. The<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1114,1102],"class_list":["post-90222","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1114","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nAn equilateral triangle is inscribed in a circle of radius 1 unit. The<\/title>\n<meta name=\"description\" content=\"An equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360\u00b0\/3 = 120\u00b0. The area of a sector of the circle corresponding to this angle is (120\u00b0\/360\u00b0) * \u03c0 * r\u00b2 = (1\/3) * \u03c0 * 1\u00b2 = \u03c0\/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1\/2) * r * r * sin(120\u00b0) = (1\/2) * 1 * 1 * (\u221a3\/2) = \u221a3\/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: \u03c0\/3 - \u221a3\/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (\u03c0\/3 - \u221a3\/4) = \u03c0 - 3\u221a3\/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment. For an equilateral triangle inscribed in a circle, each side subtends a central angle of 120\u00b0. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"An equilateral triangle is inscribed in a circle of radius 1 unit. The\" \/>\n<meta property=\"og:description\" content=\"An equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360\u00b0\/3 = 120\u00b0. The area of a sector of the circle corresponding to this angle is (120\u00b0\/360\u00b0) * \u03c0 * r\u00b2 = (1\/3) * \u03c0 * 1\u00b2 = \u03c0\/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1\/2) * r * r * sin(120\u00b0) = (1\/2) * 1 * 1 * (\u221a3\/2) = \u221a3\/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: \u03c0\/3 - \u221a3\/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (\u03c0\/3 - \u221a3\/4) = \u03c0 - 3\u221a3\/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment. For an equilateral triangle inscribed in a circle, each side subtends a central angle of 120\u00b0. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:23:23+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"An equilateral triangle is inscribed in a circle of radius 1 unit. The","description":"An equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360\u00b0\/3 = 120\u00b0. The area of a sector of the circle corresponding to this angle is (120\u00b0\/360\u00b0) * \u03c0 * r\u00b2 = (1\/3) * \u03c0 * 1\u00b2 = \u03c0\/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1\/2) * r * r * sin(120\u00b0) = (1\/2) * 1 * 1 * (\u221a3\/2) = \u221a3\/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: \u03c0\/3 - \u221a3\/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (\u03c0\/3 - \u221a3\/4) = \u03c0 - 3\u221a3\/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment. For an equilateral triangle inscribed in a circle, each side subtends a central angle of 120\u00b0. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/","og_locale":"en_US","og_type":"article","og_title":"An equilateral triangle is inscribed in a circle of radius 1 unit. The","og_description":"An equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360\u00b0\/3 = 120\u00b0. The area of a sector of the circle corresponding to this angle is (120\u00b0\/360\u00b0) * \u03c0 * r\u00b2 = (1\/3) * \u03c0 * 1\u00b2 = \u03c0\/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1\/2) * r * r * sin(120\u00b0) = (1\/2) * 1 * 1 * (\u221a3\/2) = \u221a3\/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: \u03c0\/3 - \u221a3\/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (\u03c0\/3 - \u221a3\/4) = \u03c0 - 3\u221a3\/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment. For an equilateral triangle inscribed in a circle, each side subtends a central angle of 120\u00b0. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:23:23+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/","url":"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/","name":"An equilateral triangle is inscribed in a circle of radius 1 unit. The","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:23:23+00:00","dateModified":"2025-06-01T10:23:23+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"An equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360\u00b0\/3 = 120\u00b0. The area of a sector of the circle corresponding to this angle is (120\u00b0\/360\u00b0) * \u03c0 * r\u00b2 = (1\/3) * \u03c0 * 1\u00b2 = \u03c0\/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1\/2) * r * r * sin(120\u00b0) = (1\/2) * 1 * 1 * (\u221a3\/2) = \u221a3\/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: \u03c0\/3 - \u221a3\/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (\u03c0\/3 - \u221a3\/4) = \u03c0 - 3\u221a3\/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment. For an equilateral triangle inscribed in a circle, each side subtends a central angle of 120\u00b0. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/an-equilateral-triangle-is-inscribed-in-a-circle-of-radius-1-unit-the\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"An equilateral triangle is inscribed in a circle of radius 1 unit. 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