{"id":7712,"date":"2023-03-11T12:10:00","date_gmt":"2023-03-11T06:40:00","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7712"},"modified":"2023-05-23T16:33:00","modified_gmt":"2023-05-23T11:03:00","slug":"formula-frustum-cone","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/formula-frustum-cone\/","title":{"rendered":"Formula of frustum of cone"},"content":{"rendered":"\n<p>we will check out Formula of frustum of cone in this post. we will also derive the formula so that it is easy to understand.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">What is Frustum of cone<\/h2>\n\n\n\n<p>A frustum of cone is obtained by removing the upper part of the cone<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/formula-frustrum-of-cone.png\"><img loading=\"lazy\" decoding=\"async\" width=\"492\" height=\"166\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/formula-frustrum-of-cone.png\" alt=\"Formula of frustum of cone\" class=\"wp-image-7713\" srcset=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/formula-frustrum-of-cone.png 492w, https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/formula-frustrum-of-cone-300x101.png 300w\" sizes=\"auto, (max-width: 492px) 100vw, 492px\" \/><\/a><\/figure>\n\n\n\n<p>Lets take a look at the Frustum below<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/frustrum-of-cone.png\"><img loading=\"lazy\" decoding=\"async\" width=\"235\" height=\"205\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/frustrum-of-cone.png\" alt=\"Formula of frustum of cone\" class=\"wp-image-7714\"\/><\/a><\/figure>\n\n\n\n<p>H is the height of the Frustum<br>R is the radius of the Base<br>r is the radius of the Top<br>l is the lateral height or slant height of the frustum<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Formula for Volume of the Frustrum<\/h2>\n\n\n\n<p>$V = \\frac {1}{3} \\pi H(r^2 + R^2 + rR)$<\/p>\n\n\n\n<p><strong>Derivation<\/strong><\/p>\n\n\n\n<p> Lets take the full cone from where frustum is taken<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/frustrum-of-cone-of-full-cone.png\"><img loading=\"lazy\" decoding=\"async\" width=\"231\" height=\"307\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/frustrum-of-cone-of-full-cone.png\" alt=\"Proof of Formula for Volume  frustum of cone\" class=\"wp-image-7715\" srcset=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/frustrum-of-cone-of-full-cone.png 231w, https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/frustrum-of-cone-of-full-cone-226x300.png 226w\" sizes=\"auto, (max-width: 231px) 100vw, 231px\" \/><\/a><\/figure>\n\n\n\n<p>Volume of Frustum<\/p>\n\n\n\n<p>$= \\frac {1}{3} \\pi (h+H)R^2  &#8211; \\frac {1}{3} \\pi h r^2$<\/p>\n\n\n\n<p>Now from similar triangle theorem for right triangle for small clone and big close<br>$\\frac {h}{h+H} = {r}{R}$<br>or $h = H \\frac { r}{R-r}$<\/p>\n\n\n\n<p>Substituting this in Volume <\/p>\n\n\n\n<p>$=  \\frac {1}{3} \\pi (H \\frac { r}{R-r}+H)R^2  &#8211; \\frac {1}{3} \\pi H \\frac { r}{R-r} r^2$<br>$=\\frac {1}{3} \\pi H ( \\frac {R^3}{R-r} &#8211; \\frac {r^3}{R-r})$<br>$= \\frac {1}{3} \\pi H  \\frac { R^3-r^3}{R-r}$<br>Now $a^3 -b^3 = (a-b) (a^2 + b^2 +a b)$<\/p>\n\n\n\n<p>Therefore<br>$= \\frac {1}{3} \\pi H  (r^2 + R^2 + rR)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Formula for Curved Surface Area of the Frustrum<\/h2>\n\n\n\n<p>$S = \\pi l  ( r + R)$<\/p>\n\n\n\n<p>where l is the slant height and it is given by<\/p>\n\n\n\n<p>$l= \\sqrt { H^2 + (R-r)^2}$<\/p>\n\n\n\n<p><strong>Derivation<\/strong><\/p>\n\n\n\n<p> Lets take the full cone from where frustum is taken<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/derivation-of-surface-area-frustrum-of-cone.png\"><img loading=\"lazy\" decoding=\"async\" width=\"196\" height=\"302\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/derivation-of-surface-area-frustrum-of-cone.png\" alt=\"Proof of Formula for curved surface area  frustum of cone\" class=\"wp-image-7716\" srcset=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/derivation-of-surface-area-frustrum-of-cone.png 196w, https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2023\/03\/derivation-of-surface-area-frustrum-of-cone-195x300.png 195w\" sizes=\"auto, (max-width: 196px) 100vw, 196px\" \/><\/a><\/figure>\n\n\n\n<p>Curved surface area of frustum<br>$S= \\pi R (s + l)  &#8211; \\pi r s$<\/p>\n\n\n\n<p>Now from similar triangle theorem for right triangle for small clone and big close<br>$\\frac {s}{s+l} = {r}{R}$<br>or $s = l \\frac { r}{R-r}$<\/p>\n\n\n\n<p>Therefore surface area is<\/p>\n\n\n\n<p>$= \\pi R (l \\frac { r}{R-r} + l) &#8211; \\pi r l \\frac { r}{R-r} $<br>$= \\pi l  \\frac {R^2}{R-r} &#8211; \\pi l  \\frac {r^2}{R-r}$<br>$=\\pi l \\frac { R^2 -r^2}{R-r} = \\pi l (R+r )$ <\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Formula for Total  Surface Area of the Frustrum<\/h2>\n\n\n\n<p>Formula for Total surface = Curved Surface Area + Area of Top + Area  of base<br>$= \\pi l (R+r ) + \\pi (r^2 + R^2)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved Examples<\/h2>\n\n\n\n<p><strong>Question 1<\/strong><br>The radii of the top and bottom of a bucket of slant height 45 cm are 28 cm and 7 cm, respectively. Find the  curved surface area of the bucket <br><strong>Solution<\/strong><\/p>\n\n\n\n<p>Here l=45 cm, r=7 cm and R=28 cm<br>Curved Surface area= $\\pi l(R+r) = \\pi \\times 45 (28 + 7) =\\frac {22}{7} \\times 45 \\times 35 = 4950 cm^2$<\/p>\n\n\n\n<p>I hope this article on Formula of frustum of cone helps you. <\/p>\n\n\n\n<p><strong>Related Articles<\/strong><br><a href=\"https:\/\/physicscatalyst.com\/class-10-maths-ncert-solutions.php\">Class 10 Maths<\/a> <br><a href=\"https:\/\/physicscatalyst.com\/Class10\/surface-area-volume.php\">Surface Area and Volume Class 10 Notes<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>we will check out Formula of frustum of cone in this post. we will also derive the formula so that it is easy to understand. What is Frustum of cone A frustum of cone is obtained by removing the upper part of the cone Lets take a look at the Frustum below H is the [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","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":"","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":[498],"tags":[],"class_list":["post-7712","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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What is Frustum of cone A frustum of cone is obtained by removing the upper part of the cone Lets take a look at the Frustum below H is the&hellip;","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7712","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=7712"}],"version-history":[{"count":3,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7712\/revisions"}],"predecessor-version":[{"id":8053,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7712\/revisions\/8053"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=7712"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=7712"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=7712"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}