{"id":156308,"date":"2022-03-26T01:36:46","date_gmt":"2022-03-25T20:06:46","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/frustum-of-cone\/"},"modified":"2022-12-22T21:38:10","modified_gmt":"2022-12-22T16:08:10","slug":"frustum-of-cone","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/frustum-of-cone\/","title":{"rendered":"Volume of frustum of cone"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/frustum-of-cone\/#What_is_Frustum_of_Cone\" title=\"What is Frustum of Cone?\">What is Frustum of Cone?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/frustum-of-cone\/#The_Volume_of_Frustum_of_Cone\" title=\"The Volume of Frustum of Cone\">The Volume of Frustum of Cone<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/frustum-of-cone\/#Derivation_of_Volume_of_Frustum_of_Cone\" title=\"Derivation of Volume of Frustum of Cone\">Derivation of Volume of Frustum of Cone<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/frustum-of-cone\/#Formula_to_Calculate_the_Volume_of_Frustum_of_Cone\" title=\"Formula to Calculate the Volume of Frustum of Cone\">Formula to Calculate the Volume of Frustum of Cone<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Frustum_of_Cone\"><\/span>What is Frustum of Cone?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Frustum of Cone: The <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/frustum-of-cone\/\">frustum<\/a> of a cone is the portion of the cone that is cut off by the planes perpendicular to the cone&#8217;s axis.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156307 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/frustum-of-cone.jpg\" alt=\"Frustum of Cone\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/frustum-of-cone.jpg?v=1648238803 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/frustum-of-cone-300x212.jpg?v=1648238803 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"The_Volume_of_Frustum_of_Cone\"><\/span>The Volume of Frustum of Cone<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>When a cone is cut by a plane parallel to its base, a frustum is formed. The volume of the frustum is the volume of the cone minus the volume of the cone&#8217;s two bases.<\/p>\n<p>To calculate the volume of a frustum, we need to know the height, h, of the frustum, the radius of the base, r, and the slant height, s. The volume of a cone is given by:<\/p>\n<p>V = 1\/3 * pi * r^2 * h<\/p>\n<p>The volume of the frustum is given by:<\/p>\n<p>V = 1\/3 * pi * r^2 * h &#8211; (1\/3 * pi * r^2 * h)<\/p>\n<p>The volume of a frustum of a cone depends on its slant height and radius of the upper and bottom circular part. Basically a <a href=\"https:\/\/byjus.com\/maths\/frustum-of-cone\/\">frustum of a cone<\/a> is formed when we cut a right-circular cone by a plane <a href=\"https:\/\/en.wikipedia.org\/wiki\/Parallel\">parallel<\/a> to its base into two parts. Hence, this part of the cone has its surface area and volume.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table table-bordered\">\n<tbody>\n<tr>\n<td><strong>Volume of frustum of cone = \u03c0h\/3 (r<sub>1<\/sub><sup>2<\/sup>+r<sub>2<\/sub><sup>2<\/sup>+r<sub>1<\/sub>r<sub>2<\/sub>)<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Let us learn here to derive the volume of frustum and understand the concept better by solving the problems.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Derivation_of_Volume_of_Frustum_of_Cone\"><\/span>Derivation of Volume of Frustum of Cone<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let us consider a right circular cone which cut by a plane parallel to its base as given in the below figure.<\/p>\n<p><img title=\"Volume of Frustum of a Cone\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2020\/06\/volume-of-frustum-of-a-cone.png\" alt=\"Volume of Frustum of a Cone\" \/><\/p>\n<p>Here, we can consider the frustum as the difference between the two right circular cones.<\/p>\n<p>Let,<\/p>\n<p>Height of large cone = h<\/p>\n<p>Slant height of large cone = l<\/p>\n<p>and radius = r<\/p>\n<p>Now,<\/p>\n<p>Height of smaller cone = h\u2019<\/p>\n<p>Slant height of smaller cone = l\u2019<\/p>\n<p>And radius = r\u2019<\/p>\n<p>In case of frustum,<\/p>\n<p>Height of frustum = H<\/p>\n<p>Slant height= L<\/p>\n<p>Now, we can say,<\/p>\n<p>volume of bigger cone say V<sub>1<\/sub> = \u2153 \u03c0 r<sup>2<\/sup>h<\/p>\n<p>Also, volume of smaller cone, V<sub>2<\/sub> = \u2153 \u03c0r\u2019<sup>2<\/sup>h\u2019<\/p>\n<p>Therefore,<\/p>\n<p>Volume of frustum of cone, V = V<sub>1<\/sub> \u2013 V<sub>2<\/sub><\/p>\n<p>V = \u2153 \u03c0 r<sup>2<\/sup>h \u2013 \u2153 \u03c0r\u2019<sup>2<\/sup>h\u2019<\/p>\n<p>V = \u2153 \u03c0 (r<sup>2<\/sup>h \u2013 r\u2019<sup>2<\/sup>h\u2019) \u2026\u2026\u2026\u2026\u2026(1)<\/p>\n<p>From the figure, in the \u2206OO\u2019D and \u2206OPB;<\/p>\n<p>\u2220DOO\u2019 = \u2220BOP (<em>Common Angle<\/em>)<\/p>\n<p>CD\/\/AB (Plane cutting the cone is parallel to the base)<\/p>\n<p>\u21d2\u2220O\u2019DO = \u2220PBO (<em>Corresponding Angles<\/em>)<\/p>\n<p>Thus, \u2206OO\u2019D~\u2206OPD (By <em>AA criterion of similarity<\/em>)<\/p>\n<p>As we know, by the condition of similarity, the ratio of sides of similar triangles are equal.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Formula_to_Calculate_the_Volume_of_Frustum_of_Cone\"><\/span>Formula to Calculate the Volume of Frustum of Cone<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The volume of the frustum of a cone given by the following formula:<\/p>\n<p>Volume = (1\/3) * pi * h * (r1^2 + r2^2)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is Frustum of Cone? Frustum of Cone: The frustum of a cone is the portion of the cone that [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Volume of frustum of cone","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn Frustum of Cone topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online session.","custom_permalink":"maths\/frustum-of-cone\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Volume of frustum of cone - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Learn Frustum of Cone topic of Maths in details explained by subject experts on infinitylearn.com. 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