{"id":233896,"date":"2025-06-13T14:24:01","date_gmt":"2025-06-13T14:24:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233896"},"modified":"2025-06-13T14:24:03","modified_gmt":"2025-06-13T14:24:03","slug":"find-the-area-of-a-regular-decagon-with-radius-12-cm","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/find-the-area-of-a-regular-decagon-with-radius-12-cm\/","title":{"rendered":"Find the area of a regular decagon with radius 12 cm."},"content":{"rendered":"\n

Find the area of a regular decagon with radius 12 cm.A = _<\/em><\/p>\n\n\n\n

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

Answer:<\/strong>
The area AAA of a regular decagon (10-sided polygon) with a radius (distance from center to a vertex) of 12 cm is approximately: A=10\u00d712\u00d7r2\u00d7sin\u2061(360\u221810)=10\u00d712\u00d7122\u00d7sin\u2061(36\u2218)=5\u00d7144\u00d7sin\u2061(36\u2218)\u2248720\u00d70.5878\u2248423.22 cm2A = 10 \\times \\frac{1}{2} \\times r^2 \\times \\sin\\left(\\frac{360^\\circ}{10}\\right) = 10 \\times \\frac{1}{2} \\times 12^2 \\times \\sin(36^\\circ) = 5 \\times 144 \\times \\sin(36^\\circ) \\approx 720 \\times 0.5878 \\approx \\boxed{423.22\\ \\text{cm}^2}A=10\u00d721\u200b\u00d7r2\u00d7sin(10360\u2218\u200b)=10\u00d721\u200b\u00d7122\u00d7sin(36\u2218)=5\u00d7144\u00d7sin(36\u2218)\u2248720\u00d70.5878\u2248423.22 cm2\u200b<\/p>\n\n\n\n

\n\n\n\n

Explanation (Textbook Style):<\/strong><\/p>\n\n\n\n

A regular decagon is a ten-sided polygon where all sides and angles are equal. To find the area of a regular polygon given the radius (distance from the center to any vertex), the formula derived from trigonometry is: A=12\u22c5n\u22c5r2\u22c5sin\u2061(360\u2218n)A = \\frac{1}{2} \\cdot n \\cdot r^2 \\cdot \\sin\\left(\\frac{360^\\circ}{n}\\right)A=21\u200b\u22c5n\u22c5r2\u22c5sin(n360\u2218\u200b)<\/p>\n\n\n\n

Here:<\/p>\n\n\n\n

    \n
  • nnn is the number of sides (10 for a decagon),<\/li>\n\n\n\n
  • rrr is the radius (12 cm),<\/li>\n\n\n\n
  • sin\u2061(360\u2218n)\\sin\\left(\\frac{360^\\circ}{n}\\right)sin(n360\u2218\u200b) comes from dividing the polygon into nnn congruent isosceles triangles from the center.<\/li>\n<\/ul>\n\n\n\n

    Plugging the values into the formula: A=12\u22c510\u22c5122\u22c5sin\u2061(36\u2218)A = \\frac{1}{2} \\cdot 10 \\cdot 12^2 \\cdot \\sin\\left(36^\\circ\\right)A=21\u200b\u22c510\u22c5122\u22c5sin(36\u2218)<\/p>\n\n\n\n

    Calculate 122=14412^2 = 144122=144. Then: A=5\u22c5144\u22c5sin\u2061(36\u2218)A = 5 \\cdot 144 \\cdot \\sin(36^\\circ)A=5\u22c5144\u22c5sin(36\u2218)<\/p>\n\n\n\n

    Using a calculator, sin\u2061(36\u2218)\u22480.5878\\sin(36^\\circ) \\approx 0.5878sin(36\u2218)\u22480.5878. So: A\u2248720\u22c50.5878\u2248423.22 cm2A \\approx 720 \\cdot 0.5878 \\approx 423.22\\ \\text{cm}^2A\u2248720\u22c50.5878\u2248423.22 cm2<\/p>\n\n\n\n

    Thus, the area of a regular decagon with a radius of 12 cm is approximately 423.22 square centimeters<\/strong>. This method uses trigonometry and is applicable for any regular polygon if the radius and number of sides are known.<\/p>\n\n\n\n

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

    Find the area of a regular decagon with radius 12 cm.A = _ The Correct Answer and Explanation is: Answer:The area AAA of a regular decagon (10-sided polygon) with a radius (distance from center to a vertex) of 12 cm is approximately: A=10\u00d712\u00d7r2\u00d7sin\u2061(360\u221810)=10\u00d712\u00d7122\u00d7sin\u2061(36\u2218)=5\u00d7144\u00d7sin\u2061(36\u2218)\u2248720\u00d70.5878\u2248423.22 cm2A = 10 \\times \\frac{1}{2} \\times r^2 \\times \\sin\\left(\\frac{360^\\circ}{10}\\right) = 10 \\times \\frac{1}{2} […]<\/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-233896","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233896","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=233896"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233896\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233896"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233896"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233896"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}