{"id":10234,"date":"2022-02-27T23:05:52","date_gmt":"2022-02-27T17:35:52","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10234"},"modified":"2022-02-28T23:58:50","modified_gmt":"2022-02-28T18:28:50","slug":"area-of-frustum-of-cone-formula-and-derivation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/area-of-frustum-of-cone-formula-and-derivation\/","title":{"rendered":"Area of Frustum of Cone – Formula and Derivation"},"content":{"rendered":"

Here you will learn formula for the total surface area and curved surface area of frustum of cone with derivation and examples.<\/p>\n

Let’s begin –<\/p>\n

What is Frustum of Cone ?<\/h2>\n

Frustum of a cone<\/strong> is the solid obtained after removing the upper portion of the cone by a plane parallel to its base. The lower portion is the frustum of a cone.<\/p>\n

Height<\/strong> : The perpendicular distance between the aforesaid plane and the base is called the height of the frustum.<\/p>\n

Area of a Frustum of a Cone<\/h2>\n

(i) Curved Surface Area (C.S.A) or Lateral Surface Area<\/h3>\n

C.S.A = \\(\\pi(r_1 + r_2)l\\)<\/p><\/blockquote>\n

(ii) Total Surface Area (T.S.A)<\/h3>\n

T.S.A = \\(\\pi(r_1 + r_2)l\\) + \\(\\pi{r_1}^2\\) + \\(\\pi{r_2}^2\\)<\/p><\/blockquote>\n

where \\(l\\) is the slant height of the frustum, \\(r_1\\), \\(r_2\\) are the radii of the base and the top of frustum of a cone.<\/p>\n

Note<\/strong> : Height h of the frustum is given by the relation,<\/p>\n

\\(l^2\\) = \\(h^2\\) + \\((r_1 – r_2)^2\\)<\/p><\/blockquote>\n

Also Read<\/strong> : Volume of a Frustum of a Cone \u2013 Formula and Derivation<\/a><\/p>\n

Derivation :<\/h3>\n

Let \\(r_1\\) and \\(r_2\\) be the radii of the bottom and top of the frustum respectively and \\(l\\) be the slant height of the frustum.\"area<\/p>\n

Curved Surface area of bigger cone – curved surface area of smaller cone<\/p>\n

= \\(\\pi r_1l_1\\) – \\(\\pi r_2l_2\\)<\/p>\n

Since, [ \\(l_1\\over r_1\\) = \\(l_2\\over r_2\\) = k ]<\/p>\n

= \\(\\pi[r_1(kr_1) – r_2l(kr_2)]\\)<\/p>\n

= \\(\\pi k\\) \\([{r_1}^2 – {r_2}^2]\\)<\/p>\n

C.S.A = \\(\\pi k\\)\\((r_1 – r_2)\\)\\((r_1 + r_2)\\) = \\(\\pi\\)\\((kr_1 – kr_2)\\)\\((r_1 + r_2)\\)<\/p>\n

C.S.A = \\(\\pi\\) \\((l_1 – l_2)\\)\\((r_1 + r_2)\\) = \\(\\pi l(r_1 + r_2)\\)<\/p>\n

= (semiperimeter of top and bottom) \\(\\times\\) (slant height)<\/p>\n

Total surface are of frustum = Curved surface area + area of top + area of bottom<\/p>\n

T.S.A = \\(\\pi(r_1 + r_2)l\\) + \\(\\pi{r_1}^2\\) + \\(\\pi{r_2}^2\\)<\/p>\n

Example<\/strong><\/span> : A friction clutch is in the form of the frustum of a cone, the diameters of the ends being 8 cm, and 10 cm and length 8 cm. Find its bearing Surface area.<\/p>\n

Solution<\/strong><\/span> : Here, radius,<\/p>\n

\\(r_1\\) = \\(10\\over 2\\) = 5cm,<\/p>\n

\\(r_2\\) = \\(8\\over 2\\) = 4cm<\/p>\n

Slant height, l = 8 cm<\/p>\n

Area of the bearing surface of the cone = \\(\\pi(r_1 + r_2)l\\)<\/p>\n

= 3.14(4 + 5) \\(\\times\\) 8 = 226.08 \\(cm^2\\)<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula for the total surface area and curved surface area of frustum of cone with derivation and examples. Let’s begin – What is Frustum of Cone ? Frustum of a cone is the solid obtained after removing the upper portion of the cone by a plane parallel to its base. The …<\/p>\n

Area of Frustum of Cone – Formula and Derivation<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[63],"tags":[908,904,905,907,902,903,906],"yoast_head":"\nArea of Frustum of Cone - Formula and Derivation<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula for the total surface area and curved surface area of frustum of cone with derivation and examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/area-of-frustum-of-cone-formula-and-derivation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Area of Frustum of Cone - 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