{"id":390,"date":"2022-04-13T10:51:16","date_gmt":"2022-04-13T10:51:16","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=390"},"modified":"2025-01-24T11:44:49","modified_gmt":"2025-01-24T11:44:49","slug":"t-8-surface-area-and-volume-of-similar-solids","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/t-8-surface-area-and-volume-of-similar-solids\/","title":{"rendered":"T.8 Surface area and volume of similar solids"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Surface area and volume of similar solids<\/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-15701871284c3b81cf365858c7f53f37\" style=\"color:#000060\"><strong>Definition of Similar Solids<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Two solids are <strong>similar<\/strong> if they have the same shape but different sizes.<\/li>\n\n\n\n<li>Corresponding angles are equal, and corresponding sides are proportional.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-dd2cacf3d9013d37ed9dcfc746524a37\" style=\"color:#000060\"><strong>Scale Factor (k)<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>The <strong>scale factor<\/strong> is the ratio of corresponding linear dimensions (lengths, widths, or heights) of two similar solids.<\/li>\n\n\n\n<li>If the scale factor between two similar solids is <strong>k<\/strong>, then all corresponding linear measurements (edges, radii, heights) are in the ratio <strong>1:k<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-43b7334cc199ad62c95197385f7fd5d7\" style=\"color:#000060\"><strong>Surface Area Ratio<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>The ratio of the <strong>surface areas<\/strong> of two similar solids is the <strong>square of the scale factor<\/strong>.<\/li>\n\n\n\n<li>Formula: <\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\"><strong>Surface\u00a0Area\u00a0of\u00a0Solid\u00a01 \/ Surface\u00a0Area\u00a0of\u00a0Solid\u00a02 = k<sup>2<\/sup><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>If the scale factor is <strong>k<\/strong>, then: <\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\"><strong>Surface\u00a0Area\u00a0of\u00a0larger\u00a0solid = k<sup>2<\/sup> \u00d7 Surface\u00a0Area\u00a0of\u00a0smaller\u00a0solid<\/strong><\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-9c29c2506cab34ac7da04a6b8ef0ea2f\" style=\"color:#000060\"><strong>Volume Ratio<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>The ratio of the <strong>volumes<\/strong> of two similar solids is the <strong>cube of the scale factor<\/strong>.<\/li>\n\n\n\n<li>Formula: <\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\"><strong>Volume\u00a0of\u00a0Solid\u00a01 \/ Volume\u00a0of\u00a0Solid\u00a02 = k<sup>3<\/sup><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>If the scale factor is <strong>k<\/strong>, then: <\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-large-font-size\">Volume\u00a0of\u00a0larger\u00a0solid = k<sup>3<\/sup> \u00d7 Volume\u00a0of\u00a0smaller\u00a0solid<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d933db7539329b7fc35048bdc80b8883\" style=\"color:#000060\"><strong>Example Calculation<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">If two similar spheres have a scale factor of <strong>2<\/strong>, then:<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Their <strong>surface area ratio<\/strong> is <strong>2<sup>2<\/sup> = 4<\/strong>.<\/li>\n\n\n\n<li>Their <strong>volume ratio<\/strong> is <strong>2<sup>3<\/sup> = 8<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-a63742b2848f076b934b602af78efcb3\" style=\"color:#000060\"><strong>Key Formulas Summary<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\"><strong>Scale Factor (k):<\/strong> Ratio of corresponding lengths.<\/p>\n\n\n\n<p class=\"has-large-font-size\"><strong>Surface Area Ratio:<\/strong> k<sup>2<\/sup>.<\/p>\n\n\n\n<p class=\"has-large-font-size\"><strong>Volume Ratio:<\/strong> k<sup>3<\/sup>.<\/p>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" 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:#f4b4b4\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>The figures below are similar.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"670\" height=\"300\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-15.png\" alt=\"\" class=\"wp-image-10884\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-15.png 670w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-15-300x134.png 300w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><\/figure>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-a5b27788ddfef3b24bef7b9c9d122dd7\" style=\"color:#b00012\"><strong>What is the volume of the smaller triangular prism?<\/strong><\/p>\n\n\n\n<p><em>V2<\/em>&nbsp;=&nbsp;_____&nbsp;cubic millimetres<\/p>\n<\/div><\/div>\n\n\n\n<p>Find the cube of the ratio of the corresponding dimensions:<\/p>\n\n\n\n<p>(a\/v)<sup>3 <\/sup>= (8\/6)<sup>3 <\/sup>= (4\/3)<sup>3 <\/sup>= 64\/27<\/p>\n\n\n\n<p>Find the ratio of the volumes:<\/p>\n\n\n\n<p>V1\/V2 = 64\/V2<\/p>\n\n\n\n<p>Use these two ratios to set up a proportion and solve for&nbsp;<em>V2<\/em>.<\/p>\n\n\n\n<p>64\/27 = 64\/V2<\/p>\n\n\n\n<p>64\/27 ( 27V<sub>2<\/sub> ) = 64\/V<sub>2<\/sub> ( 27V<sub>2<\/sub> )                     Multiply both sides by 27V2<\/p>\n\n\n\n<p>64V<sub>2<\/sub> = 64 \u00b7 27                    Simplify<\/p>\n\n\n\n<p>64V<sub>2<\/sub> = 1,728                    Simplify<\/p>\n\n\n\n<p>64V<sub>2<\/sub>\u00f7 64 = 1,728 \u00f7 64              Divide both sides by 64<\/p>\n\n\n\n<p>V<sub>2<\/sub>= 27<\/p>\n\n\n\n<p>The volume of the smaller triangular prism is 27 cubic millimetres.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f4aede\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>The figures below are similar.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"670\" height=\"300\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-1-4.png\" alt=\"\" class=\"wp-image-10890\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-1-4.png 670w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-1-4-300x134.png 300w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><\/figure>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-955584937cab6dad4f43eb53ce0a3ecd\" style=\"color:#b00012\"><strong>What is the surface area of the smaller cylinder?<\/strong><\/p>\n\n\n\n<p><em>S<sub>2<\/sub><\/em>&nbsp;=&nbsp;_______&nbsp;square centimetres<\/p>\n<\/div><\/div>\n\n\n\n<p>Find the square of the ratio of the corresponding dimensions:<\/p>\n\n\n\n<p>(a\/b)<sup>2 <\/sup>= (4\/2)<sup>2 <\/sup>= (2\/1)<sup>1 <\/sup>= 4\/1<\/p>\n\n\n\n<p>Find the ratio of the surface areas:<\/p>\n\n\n\n<p>S<sub>1<\/sub>\/S<sub>2<\/sub> = 196\/S<sub>2<\/sub><\/p>\n\n\n\n<p>Use these two ratios to set up a proportion and solve for&nbsp;<em><strong>S<\/strong><\/em><sub><strong>2<\/strong><\/sub>.<\/p>\n\n\n\n<p>4\/1 = 196\/S<sub>2<\/sub><\/p>\n\n\n\n<p>4\/1 ( S<sub>2<\/sub> ) = 196\/S<sub>2<\/sub> ( S<sub>2<\/sub> )<\/p>\n\n\n\n<p>4S<sub>2<\/sub>=196 \u00b7 1               Simplify<\/p>\n\n\n\n<p>4S<sub>2<\/sub>= 196               Simplify<\/p>\n\n\n\n<p>4S<sub>2<\/sub>\u00f7 4=196 \u00f7 4                Divide both sides by 4<\/p>\n\n\n\n<p>S<sub>2<\/sub>= 49<\/p>\n\n\n\n<p>The surface area of the smaller cylinder is 49 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\/84787\/336\/516\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-139.png\" alt=\"\" class=\"wp-image-7415\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-139.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-139-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-139-150x150.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\/84787\/211\/168\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-158.png\" alt=\"\" class=\"wp-image-7416\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-158.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-158-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-158-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Surface area and volume of similar solids key notes : Definition of Similar Solids Scale Factor (k) Surface Area Ratio Surface\u00a0Area\u00a0of\u00a0Solid\u00a01 \/ Surface\u00a0Area\u00a0of\u00a0Solid\u00a02 = k2 Surface\u00a0Area\u00a0of\u00a0larger\u00a0solid = k2 \u00d7 Surface\u00a0Area\u00a0of\u00a0smaller\u00a0solid Volume Ratio Volume\u00a0of\u00a0Solid\u00a01 \/ Volume\u00a0of\u00a0Solid\u00a02 = k3 Volume\u00a0of\u00a0larger\u00a0solid = k3 \u00d7 Volume\u00a0of\u00a0smaller\u00a0solid Example Calculation If two similar spheres have a scale factor of 2, then:<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/t-8-surface-area-and-volume-of-similar-solids\/\">Continue reading <span class=\"screen-reader-text\">&#8220;T.8 Surface area and volume of similar solids&#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-390","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/390","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=390"}],"version-history":[{"count":13,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/390\/revisions"}],"predecessor-version":[{"id":17476,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/390\/revisions\/17476"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=390"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}