{"id":182,"date":"2022-04-13T08:38:44","date_gmt":"2022-04-13T08:38:44","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=182"},"modified":"2025-02-12T10:41:16","modified_gmt":"2025-02-12T10:41:16","slug":"k-3-surface-area-of-cones","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/k-3-surface-area-of-cones\/","title":{"rendered":"K.3 Surface area of cones"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Surface area of cones<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-203290aba75681a4fb07a0852f132b50\" style=\"color:#74008b;text-transform:capitalize\">key notes :<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-8f0e9d9c5daaa1c7c095d7e98da4d9b6\" style=\"color:#000060\"><strong>Definition of a Cone<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"has-large-font-size\">A cone is a three-dimensional geometric shape with a circular base and a curved surface that tapers to a single point (apex).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-f237124a1f7665c799a947fb05e7de2b\" style=\"color:#000060\"><strong>Parts of a Cone<\/strong><\/p>\n\n\n\n<ul class=\"has-large-font-size wp-block-list\">\n<li><strong>Radius (r):<\/strong> The distance from the center to the edge of the base.<\/li>\n\n\n\n<li><strong>Height (h):<\/strong> The perpendicular distance from the base to the apex.<\/li>\n\n\n\n<li><strong>Slant height (l):<\/strong> The distance from any point on the edge of the base to the apex.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-107b78a533885ba3d58f616cf9d424da\" style=\"color:#000060\"><strong>Formula for Surface Area of a Cone<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">The <strong>total surface area (TSA)<\/strong> of a cone includes both the curved surface and the circular base: <\/p>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">TSA = \u03c0rl + \u03c0r<sup>2<\/sup> <\/p>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\"> TSA = \u03c0r (l + r) <\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-ca28fba281bade8f382fcecbbdcaa98c\" style=\"color:#000060\"><strong>where:<\/strong><\/p>\n\n\n\n<ul class=\"has-large-font-size wp-block-list\">\n<li>\u03c0 \u2248 3.1416<\/li>\n\n\n\n<li>r = radius<\/li>\n\n\n\n<li>l = slant height<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-b1c5bd1da83dc6b8194f14b80220a72e\" style=\"color:#000060\"><strong>Curved Surface Area (CSA)<\/strong><\/p>\n\n\n\n<ul class=\"has-large-font-size wp-block-list\">\n<li>The curved (lateral) surface area of a cone is given by: <\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\"><strong>CSA = \u03c0rl<\/strong><\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-f1917174654b68988ba5077e01bd9354\" style=\"color:#000060\"><strong>Finding the Slant Height (If Not Given)<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"has-large-font-size\">If the slant height is not given, it can be calculated using the <strong>Pythagorean theorem<\/strong>: <\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">l = \u221ar<sup>2<\/sup>+h<sup>2 <\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">where h is the vertical height of the cone.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-c815a6ce1ee52650e1a2f8b6574554d2\" style=\"color:#000060\"><strong>Units<\/strong><\/p>\n\n\n\n<ul class=\"has-large-font-size wp-block-list\">\n<li>The surface area is measured in <strong>square units<\/strong> (e.g., cm\u00b2, m\u00b2).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d5fa5f08e8c24845f0de10f7dfb00330\" style=\"color:#000060\"><strong>Applications of Surface Area of a Cone<\/strong><\/p>\n\n\n\n<ul class=\"has-large-font-size wp-block-list\">\n<li>Used in designing party hats, ice cream cones, traffic cones, and water tanks.<\/li>\n\n\n\n<li>Important in real-life constructions like rooftops and tents.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-76f45242e8bf94cf518c1566b523a851\" style=\"color:#000060\"><strong>Example Problem<\/strong><\/p>\n\n\n\n<ul class=\"has-large-font-size wp-block-list\">\n<li>If a cone has a radius of <strong>5 cm<\/strong> and a slant height of <strong>12 cm<\/strong>, find its total surface area.<\/li>\n\n\n\n<li><strong>Solution:<\/strong> <\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">CSA = \u03c0(5)(12) = 60\u03c0 = 188.4\u00a0cm<sup>2<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">Base\u00a0Area = \u03c0(5)<sup>2 <\/sup>= 25\u03c0 = 78.5\u00a0cm<sup>2 <\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">TSA = 188.4 + 78.5 = 266.9\u00a0cm<sup>2<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center has-text-color has-link-color has-large-font-size wp-elements-3fefd20f8ce119a518e5f53333ab5486\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f9dcc0\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>What is the surface area of this cone?<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-aceb31132f26cef0978b33376694437e\" style=\"color:#b00012\"><strong>Use&nbsp;\u200b\ud835\udf0b&nbsp;\u2248 3.14 and round your answer to the nearest hundredth.<\/strong><\/p>\n\n\n<div class=\"wp-block-image is-resized\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-8-2.png\" alt=\"\" class=\"wp-image-10819\" style=\"aspect-ratio:1;width:307px;height:auto\"\/><\/figure><\/div>\n\n\n<p>______ &nbsp;square metres<\/p>\n<\/div><\/div>\n\n\n\n<p>Find the radius and slant height of the cone.<\/p>\n\n\n\n<p>radius=1\/2 . diameter=1\/2 . 6=3<\/p>\n\n\n\n<p>slant height=8<\/p>\n\n\n\n<p>Find the area of the base. Use&nbsp;\u200b\ud835\udf0b&nbsp;\u2248 3.14.<br>Area of base&nbsp;=&nbsp;\u200b\ud835\udf0b<em>r<\/em><sup>2<\/sup><br>&nbsp;                   \u2248&nbsp;3.14 \u00d7&nbsp;3<sup>2<\/sup><br>                    &nbsp;\u2248&nbsp;28.26 <\/p>\n\n\n\n<p>Find the lateral area.<br>Lateral area&nbsp;=&nbsp;\u200b\ud835\udf0b<em>rs<\/em><br>&nbsp;                   \u2248&nbsp;3.14 \u00d7 3 \u00d7 8<br>&nbsp;                   \u2248&nbsp;75.36<\/p>\n\n\n\n<p>Now add the areas to find the surface area of the cone.<br>Surface area&nbsp;=&nbsp;area of base + lateral area<br>&nbsp;                   \u2248&nbsp;28.26 + 75.36<br>&nbsp;                  \u2248&nbsp;103.62<\/p>\n\n\n\n<p>The surface area of the cone is about 103.62 square metres.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#bdfbf7\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>What is the surface area of this cone?<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-aceb31132f26cef0978b33376694437e\" style=\"color:#b00012\"><strong>Use&nbsp;\u200b\ud835\udf0b&nbsp;\u2248 3.14 and round your answer to the nearest hundredth.<\/strong><\/p>\n\n\n<div class=\"wp-block-image is-resized\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-9-3.png\" alt=\"\" class=\"wp-image-10820\" style=\"aspect-ratio:1;width:254px;height:auto\"\/><\/figure><\/div>\n\n\n<p>_______ square metres.<\/p>\n<\/div><\/div>\n\n\n\n<p>Find the radius and slant height of the cone.<\/p>\n\n\n\n<p>radius=1\/2 . diameter=1\/2 . 2=1<\/p>\n\n\n\n<p>slant height=3<\/p>\n\n\n\n<p>Find the area of the base. Use&nbsp;\u200b\ud835\udf0b&nbsp;\u2248 3.14.<br>Area of base&nbsp;=&nbsp;\u200b\ud835\udf0b<em>r<\/em><sup>2<\/sup><br>&nbsp;                   \u2248&nbsp;3.14 \u00d7&nbsp;1<sup>2<\/sup><br>&nbsp;                   \u2248&nbsp;3.14<\/p>\n\n\n\n<p>Find the lateral area.<br>Lateral area&nbsp;=&nbsp;\u200b\ud835\udf0b<em>r<\/em><em>s<\/em><br>&nbsp;                 \u2248&nbsp;3.14 \u00d7 1 \u00d7 3<br>&nbsp;                 \u2248&nbsp;9.42<\/p>\n\n\n\n<p>Now add the areas to find the surface area of the cone.<br>Surface area&nbsp;=&nbsp;area of base + lateral area<br>&nbsp;                   \u2248&nbsp;3.14 + 9.42<br>                   &nbsp;\u2248&nbsp;12.56<\/p>\n\n\n\n<p>The surface area of the cone is about 12.56 square metres.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#feb0b0\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>What is the surface area of this cone?<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-aceb31132f26cef0978b33376694437e\" style=\"color:#b00012\"><strong>Use&nbsp;\u200b\ud835\udf0b&nbsp;\u2248 3.14 and round your answer to the nearest hundredth.<\/strong><\/p>\n\n\n<div class=\"wp-block-image is-resized\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-10-1.png\" alt=\"\" class=\"wp-image-10821\" style=\"aspect-ratio:1;width:285px;height:auto\"\/><\/figure><\/div>\n\n\n<p>_____square centimetres<\/p>\n<\/div><\/div>\n\n\n\n<p>Find the radius and slant height of the cone.<\/p>\n\n\n\n<p>radius&nbsp;=&nbsp;5<br>slant height&nbsp;=&nbsp;10<\/p>\n\n\n\n<p><br>Find the area of the base. Use&nbsp;\u200b\ud835\udf0b&nbsp;\u2248 3.14.<br>Area of base&nbsp;=&nbsp;\u200b\ud835\udf0b<em>r<\/em><sup>2<\/sup><br>&nbsp;                    \u2248&nbsp;3.14 \u00d7&nbsp;5<sup>2<\/sup><br>&nbsp;                   \u2248&nbsp;78.5<\/p>\n\n\n\n<p>Find the lateral area.<br>Lateral area&nbsp;=&nbsp;\u200b\ud835\udf0b<em>r<\/em><em>s<\/em><br>&nbsp;                   \u2248&nbsp;3.14 \u00d7 5 \u00d7 10<br>                 &nbsp;\u2248&nbsp;157<\/p>\n\n\n\n<p>Now add the areas to find the surface area of the cone.<br>Surface area&nbsp;=&nbsp;area of base + lateral area<br>&nbsp;                  \u2248&nbsp;78.5 + 157<br>&nbsp;                 \u2248&nbsp;235.5<\/p>\n\n\n\n<p>The surface area of the cone is about 235.50 square centimetres.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\">let&#8217;s practice!<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/86042\/695\/774\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-65.png\" alt=\"\" class=\"wp-image-7880\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-65.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-65-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-65-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/86042\/687\/172\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-74.png\" alt=\"\" class=\"wp-image-7881\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-74.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-74-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-74-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Surface area of cones key notes : Definition of a Cone Parts of a Cone Formula for Surface Area of a Cone The total surface area (TSA) of a cone includes both the curved surface and the circular base: TSA = \u03c0rl + \u03c0r2 TSA = \u03c0r (l + r) where: Curved Surface Area (CSA)<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/k-3-surface-area-of-cones\/\">Continue reading <span class=\"screen-reader-text\">&#8220;K.3 Surface area of cones&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-182","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=182"}],"version-history":[{"count":20,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/182\/revisions"}],"predecessor-version":[{"id":16516,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/182\/revisions\/16516"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}