{"id":185566,"date":"2025-01-22T18:09:15","date_gmt":"2025-01-22T18:09:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=185566"},"modified":"2025-01-22T18:09:18","modified_gmt":"2025-01-22T18:09:18","slug":"find-the-sum-of-the-interior-angles-of-a-decagon","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/22\/find-the-sum-of-the-interior-angles-of-a-decagon\/","title":{"rendered":"Find the sum of the interior angles of a decagon"},"content":{"rendered":"\n
    \n
  1. Find the sum of the interior angles of a decagon. 2.If the exterior angle of a regular polygon is 30\u00c3\u201a\u00c2\u00b0, how many sides does the polygon have? 3.A regular polygon has an interior angle of 160\u00c3\u201a\u00c2\u00b0.How many sides does it have? 4.If the sum of the interior angles of a regular polygon is 1080\u00c3\u201a\u00c2\u00b0, how many sides does the polygon have?<\/li>\n\n\n\n
  2. Find the sum of the interior angles of a decagon.<\/li>\n<\/ol>\n\n\n\n

    2.If the exterior angle of a regular polygon is 30\u00c3\u201a\u00c2\u00b0, how many sides does the polygon have?<\/p>\n\n\n\n

    3.A regular polygon has an interior angle of 160\u00c3\u201a\u00c2\u00b0.How many sides does it have?<\/p>\n\n\n\n

    4.If the sum of the interior angles of a regular polygon is 1080\u00c3\u201a\u00c2\u00b0, how many sides does the polygon have?<\/p>\n\n\n\n

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

    Let’s break down and answer each of the four questions one by one.<\/p>\n\n\n\n

    1. Sum of the Interior Angles of a Decagon<\/strong><\/h3>\n\n\n\n

    A decagon has 10 sides. To find the sum of the interior angles of any polygon, we use the formula:<\/p>\n\n\n\n

    [
    \\text{Sum of interior angles} = (n – 2) \\times 180^\\circ
    ]<\/p>\n\n\n\n

    where (n) is the number of sides. For a decagon ((n = 10)):<\/p>\n\n\n\n

    [
    \\text{Sum of interior angles} = (10 – 2) \\times 180^\\circ = 8 \\times 180^\\circ = 1440^\\circ
    ]<\/p>\n\n\n\n

    So, the sum of the interior angles of a decagon is 1440\u00b0<\/strong>.<\/p>\n\n\n\n

    2. If the Exterior Angle of a Regular Polygon is 30\u00b0<\/strong><\/h3>\n\n\n\n

    For any regular polygon, the exterior angle is given by:<\/p>\n\n\n\n

    [
    \\text{Exterior angle} = \\frac{360^\\circ}{n}
    ]<\/p>\n\n\n\n

    where (n) is the number of sides. If the exterior angle is (30^\\circ), we set up the equation:<\/p>\n\n\n\n

    [
    30^\\circ = \\frac{360^\\circ}{n}
    ]<\/p>\n\n\n\n

    Solving for (n):<\/p>\n\n\n\n

    [
    n = \\frac{360^\\circ}{30^\\circ} = 12
    ]<\/p>\n\n\n\n

    Thus, the polygon has 12 sides<\/strong>.<\/p>\n\n\n\n

    3. If the Interior Angle of a Regular Polygon is 160\u00b0<\/strong><\/h3>\n\n\n\n

    The formula for the interior angle of a regular polygon is:<\/p>\n\n\n\n

    [
    \\text{Interior angle} = \\frac{(n – 2) \\times 180^\\circ}{n}
    ]<\/p>\n\n\n\n

    We are given that the interior angle is (160^\\circ). Setting up the equation:<\/p>\n\n\n\n

    [
    160^\\circ = \\frac{(n – 2) \\times 180^\\circ}{n}
    ]<\/p>\n\n\n\n

    Multiplying both sides by (n):<\/p>\n\n\n\n

    [
    160n = (n – 2) \\times 180
    ]<\/p>\n\n\n\n

    Expanding:<\/p>\n\n\n\n

    [
    160n = 180n – 360
    ]<\/p>\n\n\n\n

    Rearranging the terms:<\/p>\n\n\n\n

    [
    180n – 160n = 360
    ]<\/p>\n\n\n\n

    [
    20n = 360
    ]<\/p>\n\n\n\n

    Solving for (n):<\/p>\n\n\n\n

    [
    n = \\frac{360}{20} = 18
    ]<\/p>\n\n\n\n

    Thus, the polygon has 18 sides<\/strong>.<\/p>\n\n\n\n

    4. If the Sum of the Interior Angles of a Regular Polygon is 1080\u00b0<\/strong><\/h3>\n\n\n\n

    The sum of the interior angles of a regular polygon is given by:<\/p>\n\n\n\n

    [
    \\text{Sum of interior angles} = (n – 2) \\times 180^\\circ
    ]<\/p>\n\n\n\n

    We are given that the sum is 1080\u00b0. Setting up the equation:<\/p>\n\n\n\n

    [
    1080^\\circ = (n – 2) \\times 180^\\circ
    ]<\/p>\n\n\n\n

    Dividing both sides by 180\u00b0:<\/p>\n\n\n\n

    [
    \\frac{1080^\\circ}{180^\\circ} = n – 2
    ]<\/p>\n\n\n\n

    [
    6 = n – 2
    ]<\/p>\n\n\n\n

    Solving for (n):<\/p>\n\n\n\n

    [
    n = 6 + 2 = 8
    ]<\/p>\n\n\n\n

    Thus, the polygon has 8 sides<\/strong>.<\/p>\n\n\n\n

    Summary of Answers:<\/h3>\n\n\n\n
      \n
    1. The sum of the interior angles of a decagon is 1440\u00b0<\/strong>.<\/li>\n\n\n\n
    2. A regular polygon with an exterior angle of 30\u00b0 has 12 sides<\/strong>.<\/li>\n\n\n\n
    3. A regular polygon with an interior angle of 160\u00b0 has 18 sides<\/strong>.<\/li>\n\n\n\n
    4. A regular polygon with a sum of interior angles of 1080\u00b0 has 8 sides<\/strong>.<\/li>\n<\/ol>\n\n\n\n

      These problems involve applying formulas for polygons, which are essential for understanding the geometric properties of various shapes.<\/p>\n","protected":false},"excerpt":{"rendered":"

      2.If the exterior angle of a regular polygon is 30\u00c3\u201a\u00c2\u00b0, how many sides does the polygon have? 3.A regular polygon has an interior angle of 160\u00c3\u201a\u00c2\u00b0.How many sides does it have? 4.If the sum of the interior angles of a regular polygon is 1080\u00c3\u201a\u00c2\u00b0, how many sides does the polygon have? The Correct Answer and […]<\/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-185566","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185566","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=185566"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185566\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185566"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185566"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}