{"id":233527,"date":"2025-06-13T07:45:46","date_gmt":"2025-06-13T07:45:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233527"},"modified":"2025-06-13T07:45:48","modified_gmt":"2025-06-13T07:45:48","slug":"regular-icosagon-and-a-regular-tetracontagon-are-inscribed-in-a-circle-with-a-radius-of-15-units-which-of-the-two-polygons-has-bigger-area-and-how-much","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/regular-icosagon-and-a-regular-tetracontagon-are-inscribed-in-a-circle-with-a-radius-of-15-units-which-of-the-two-polygons-has-bigger-area-and-how-much\/","title":{"rendered":"Regular icosagon and a regular tetracontagon are inscribed in a circle with a radius of 15 units Which of the two polygons has bigger area and how much"},"content":{"rendered":"\n<p>Regular icosagon and a regular tetracontagon are inscribed in a circle with a radius of 15 units Which of the two polygons has bigger area and how much? Tetracontagon with A = 695.28 sq.units Icosagon with A = 703.96 sq.units Icosagon with A = 695.28 sq.units<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine which of the two polygons\u2014a <strong>regular icosagon (20 sides)<\/strong> and a <strong>regular tetracontagon (40 sides)<\/strong>\u2014has a <strong>larger area<\/strong> when inscribed in a circle of radius <strong>15 units<\/strong>, we apply the <strong>standard formula<\/strong> for the area AAA of a regular nnn-gon inscribed in a circle of radius rrr:A=n2r2sin\u2061(2\u03c0n)A = \\frac{n}{2} r^2 \\sin\\left(\\frac{2\\pi}{n}\\right)A=2n\u200br2sin(n2\u03c0\u200b)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 1: Calculate the area of the icosagon (n = 20)<\/strong><\/h3>\n\n\n\n<p>A20=202\u22c5152\u22c5sin\u2061(2\u03c020)=10\u22c5225\u22c5sin\u2061(\u03c05)=2250\u22c5sin\u2061(36\u2218)A_{20} = \\frac{20}{2} \\cdot 15^2 \\cdot \\sin\\left(\\frac{2\\pi}{20}\\right) = 10 \\cdot 225 \\cdot \\sin\\left(\\frac{\\pi}{5}\\right) = 2250 \\cdot \\sin\\left(36^\\circ\\right)A20\u200b=220\u200b\u22c5152\u22c5sin(202\u03c0\u200b)=10\u22c5225\u22c5sin(5\u03c0\u200b)=2250\u22c5sin(36\u2218)sin\u2061(36\u2218)\u22480.5878\u21d2A20\u22482250\u22c50.5878\u22481322.55&nbsp;square&nbsp;units\\sin(36^\\circ) \\approx 0.5878 \\Rightarrow A_{20} \\approx 2250 \\cdot 0.5878 \\approx 1322.55 \\text{ square units}sin(36\u2218)\u22480.5878\u21d2A20\u200b\u22482250\u22c50.5878\u22481322.55&nbsp;square&nbsp;units<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 2: Calculate the area of the tetracontagon (n = 40)<\/strong><\/h3>\n\n\n\n<p>A40=402\u22c5152\u22c5sin\u2061(2\u03c040)=20\u22c5225\u22c5sin\u2061(\u03c010)=4500\u22c5sin\u2061(18\u2218)A_{40} = \\frac{40}{2} \\cdot 15^2 \\cdot \\sin\\left(\\frac{2\\pi}{40}\\right) = 20 \\cdot 225 \\cdot \\sin\\left(\\frac{\\pi}{10}\\right) = 4500 \\cdot \\sin(18^\\circ)A40\u200b=240\u200b\u22c5152\u22c5sin(402\u03c0\u200b)=20\u22c5225\u22c5sin(10\u03c0\u200b)=4500\u22c5sin(18\u2218)sin\u2061(18\u2218)\u22480.3090\u21d2A40\u22484500\u22c50.3090\u22481390.5&nbsp;square&nbsp;units\\sin(18^\\circ) \\approx 0.3090 \\Rightarrow A_{40} \\approx 4500 \\cdot 0.3090 \\approx 1390.5 \\text{ square units}sin(18\u2218)\u22480.3090\u21d2A40\u200b\u22484500\u22c50.3090\u22481390.5&nbsp;square&nbsp;units<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Comparison and Conclusion<\/strong><\/h3>\n\n\n\n<p>Comparing the two:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Area of icosagon \u2248 1322.55 sq. units<\/strong><\/li>\n\n\n\n<li><strong>Area of tetracontagon \u2248 1390.5 sq. units<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Thus, the <strong>regular tetracontagon<\/strong> has the <strong>larger area<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Correct Answer:<\/strong><\/h3>\n\n\n\n<p><strong>Tetracontagon with A = 1390.5 sq. units<\/strong><\/p>\n\n\n\n<p>This result follows a geometric principle: as the number of sides of a regular polygon inscribed in a circle increases, the polygon more closely approximates the area of the circle itself. Therefore, the <strong>more sides<\/strong>, the <strong>larger the area<\/strong>, given the same radius.<\/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-banner8-262.jpeg\" alt=\"\" class=\"wp-image-233528\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Regular icosagon and a regular tetracontagon are inscribed in a circle with a radius of 15 units Which of the two polygons has bigger area and how much? Tetracontagon with A = 695.28 sq.units Icosagon with A = 703.96 sq.units Icosagon with A = 695.28 sq.units The Correct Answer and Explanation is: To determine which [&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-233527","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233527","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=233527"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233527\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233527"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233527"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233527"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}