{"id":10232,"date":"2022-02-27T23:02:51","date_gmt":"2022-02-27T17:32:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10232"},"modified":"2022-03-01T00:00:44","modified_gmt":"2022-02-28T18:30:44","slug":"volume-of-a-frustum-of-a-cone-formula-and-derivation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/volume-of-a-frustum-of-a-cone-formula-and-derivation\/","title":{"rendered":"Volume of a Frustum of a Cone – Formula and Derivation"},"content":{"rendered":"

Here you will learn formula for the volume of a frustum of a cone with derivation and examples based on it.<\/p>\n

Let’s begin –\u00a0<\/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

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

The formula for volume of a frustum of a cone is<\/p>\n

\n

V = \\(h\\over 3\\)\\([A_1 + \\sqrt{A_1A_2} + A_2]\\)<\/p>\n<\/blockquote>\n

[\\(A_1\\), \\(A_2\\) are the areas of bottom and top of the frustum]<\/p>\n

\n

V = \\(\\pi h\\over 3\\)\\([{r_1}^2 + \\sqrt{r_1r_2} + {r_2}^2]\\)<\/p>\n<\/blockquote>\n

where h is the 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

\n

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

Also Read<\/strong> : Area 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 h be the height of the frustum.\"volume<\/p>\n

The frustum of a cone is the solid obtained after removing the upper portion (small cone) of it by a plane parallel to the base of the big cone.<\/p>\n

Volume of frustum = Volume of bigger cone – Volume of smaller cone<\/p>\n

= \\(1\\over 3\\)\\(\\pi {r_1}^2 h_1\\) – \\(1\\over 3\\)\\(\\pi {r_2}^2 h_2\\)<\/p>\n

Since, [ \\(h_1\\over r_1\\) = \\(h_2\\over r_2\\) = k ]<\/p>\n

= \\(\\pi k\\over 3\\)\\([{r_1}^3 – {r_2}^3]\\)<\/p>\n

= \\(\\pi k\\over 3\\) (\\(r_1 – r_2\\)) \\(({r_1}^2 + {r_1r_2} + {r_2}^2)\\)<\/p>\n

Volume = \\(1\\over 3\\) (\\(kr_1 – kr_2\\)) \\((\\pi{r_1}^2 + \\pi{r_1r_2} + \\pi{r_2}^2)\\)<\/p>\n

V = \\(1\\over 3\\) (\\(h_1 – h_2\\)) \\((\\pi{r_1}^2 + \\sqrt{(\\pi{r_1}^2)(\\pi{r_2}^2)} + \\pi{r_2}^2)\\)<\/p>\n

Volume = \\(h\\over 3\\)\\([A_1 + \\sqrt{A_1A_2} + A_2]\\)<\/p>\n

Where \\(A_1\\), \\(A_2\\) are the areas of bottom and top of the frustum respectively and h = \\(h_1 – h_2\\).<\/p>\n

Volume of frustum of Cone = \\(h\\over 3\\)\\([A_1 + \\sqrt{A_1A_2} + A_2]\\)<\/p>\n

= \\(h\\over 3\\) \\((\\pi{r_1}^2 + \\sqrt{(\\pi{r_1}^2)(\\pi{r_2}^2)} + \\pi{r_2}^2)\\)<\/p>\n

Volume\u00a0 = \\(\\pi h\\over 3\\)\\([{r_1}^2 + \\sqrt{r_1r_2} + {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 Volume.<\/p>\n

Solution<\/strong><\/span> : Here, radius,\u00a0<\/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

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

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

\\(\\implies\\)\u00a0 64 = \\(h^2\\) + \\((5 – 4)^2\\)\u00a0 or\u00a0 \\(h^2\\) = 63\u00a0 \\(\\implies\\) h = 7.937<\/p>\n

\\(\\therefore\\)\u00a0 Volume = \\(\\pi h\\over 3\\)\\([{r_1}^2 + \\sqrt{r_1r_2} + {r_2}^2]\\)<\/p>\n

Volume = \\(3.14 \\times 7.937\\over 3\\) \\(\\times\\) [25 + 20 + 16] = 506.75 \\(cm^2\\)<\/p>\n\n\n

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

Here you will learn formula for the volume of a frustum of a cone with derivation and examples based on it. Let’s begin –\u00a0 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 lower …<\/p>\n

Volume of a Frustum of a 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":[900,901,899],"yoast_head":"\nVolume of a Frustum of a Cone - Formula and Derivation<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula for the volume of a frustum of a cone with derivation and examples based on it.\" \/>\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\/volume-of-a-frustum-of-a-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=\"Volume of a Frustum of a Cone - 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