{"id":13252,"date":"2014-02-07T19:58:06","date_gmt":"2014-02-07T19:58:06","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=13252"},"modified":"2026-03-26T09:17:56","modified_gmt":"2026-03-26T14:17:56","slug":"proportional-change-of-dimensions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/proportional-change-of-dimensions\/","title":{"rendered":"Proportional Change of Dimensions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_PWMHYjkO3OI\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_PWMHYjkO3OI\" data-source-videoID=\"PWMHYjkO3OI\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Proportional Change of Dimensions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Proportional Change of Dimensions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_PWMHYjkO3OI:hover {cursor:pointer;} img#videoThumbnailImage_PWMHYjkO3OI {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1303-proportional-change-of-dimensions-copy-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_PWMHYjkO3OI\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_PWMHYjkO3OI\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Proportional Change of Dimensions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_PWMHYjkO3OI\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_PWMHYjkO3OI\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_PWMHYjkO3OI\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction ovp_Function() {\n  var x = document.getElementById(\"ovp\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"ovp_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"ovp\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Similar_Figures\" class=\"smooth-scroll\">Similar Figures<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Proportional_Change_and_Area\" class=\"smooth-scroll\">Proportional Change and Area<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Proportional_Change_and_Volume\" class=\"smooth-scroll\">Proportional Change and Volume<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Change_of_Dimension_Practice_Questions\" class=\"smooth-scroll\">Change of Dimension Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about proportional change of dimension!<\/p>\n<p>In this video, we will explore what it means to proportionally change dimension, how proportionally changing dimension affects measures of area in 2D figures, and how proportionally changing dimension affects measures of volume in <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/3d-geometric-shapes\/\">3D solids<\/a>.<\/p>\n<h2><span id=\"Similar_Figures\" class=\"m-toc-anchor\"><\/span>Similar Figures<\/h2>\n<p>\nFirst, let\u2019s recall what it means for two figures that are different sizes to be similar: They have the same shape, but their dimensions are proportional, which means they are different by a scale factor.<\/p>\n<p>Consider these squares. Their side lengths differ by a scale factor of 2 since each side of Square B is twice as long as each side of Square A. So we can say that these figures are similar.<\/p>\n<p>By contrast, these rectangles are not similar. The one on the right is twice as long as the one on the left, but their heights are the same. All lengths must differ by the same scale factor for the two figures to be proportional.<\/p>\n<h2><span id=\"Proportional_Change_and_Area\" class=\"m-toc-anchor\"><\/span>Proportional Change and Area<\/h2>\n<p>\nThe question we are exploring is: \u201cWhat happens to area and volume of similar figures when their dimensions change proportionally?\u201d Well, let\u2019s answer this by jumping back to our squares.<\/p>\n<p>The area of Square A is 1 and the area of Square B is 4. Visually, this is what happens:<\/p>\n<p>As you can see, four Square As can fit inside one Square B. In other words, even though Square B has 2 times the length and height of Square A, it has four times the area of Square A, not two times. Let\u2019s look at Square C, with side length 3:<\/p>\n<p>In this relationship, Square C contains nine Square As:<\/p>\n<p>Now, there is actually a pattern to be found here. As we saw with Square A and Square B, the side lengths differed by a factor of 2 and the area differed by a factor of 4. With Square A and Square C, the lengths differed by a factor of 3 and the area differed by a factor of 9.<\/p>\n<p>Now, suppose one rectangle has area 6m<sup>2<\/sup> and a similar rectangle has area 54m<sup>2<\/sup>. How much longer is each side of the large rectangle than the small one?<\/p>\n<ul>\n<li>54 = (scale factor)<sup>2<\/sup> \\(\\times\\) 6<\/li>\n<li>9 = (scale factor)<sup>2<\/sup><\/li>\n<li>3 = scale factor<\/li>\n<\/ul>\n<p>The sides of the large rectangle are 3 times the length of those of the small rectangle.<\/p>\n<h2><span id=\"Proportional_Change_and_Volume\" class=\"m-toc-anchor\"><\/span>Proportional Change and Volume<\/h2>\n<p>\nWhen we move from 2 to 3 dimensions, the effects are, well, similar! Let\u2019s look at a quick example:<\/p>\n<p>The large cube has edge lengths twice as long as the small cube.<\/p>\n<p>But 8 small cubes fit inside the large cube. Since we\u2019re talking about volume, we\u2019re going to cube instead of square, so the volume of the large cube is 2<sup>3<\/sup> times the volume of the small cube. Otherwise, the big idea is the same as with area: the volume of a proportional figure equals (scale factor)<sup>3<\/sup> times the volume of the original figure.<\/p>\n<p>And that\u2019s all there is to it! I hope this review was helpful. Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Change_of_Dimension_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Change of Dimension Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDetermine the scale factor that takes us from Rectangle X to Rectangle Y.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Proportional-Change-of-Dimensions-Example-1.svg\" alt=\"Two labeled rectangles: Rectangle X is 6 inches by 4 inches; Rectangle Y is 18 inches by 12 inches.\" width=\"516.8\" height=\"161.6\" class=\"aligncenter size-full wp-image-287309\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\"><span style=\"font-size: 120%\">\\(\\frac{1}{4}\\)<\/span><\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">3<\/div><div class=\"PQ\"  id=\"PQ-1-3\">5<\/div><div class=\"PQ\"  id=\"PQ-1-4\"><span style=\"font-size: 120%\">\\(\\frac{1}{2}\\)<\/span><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The <strong>scale factor<\/strong> is the ratio between corresponding measurements of an object. If the scale factor is a whole number, the copy will be larger. If the scale factor is a fraction, the copy will be smaller.<\/p>\n<p>The ratio of the heights is 4:12, which means that the height of Rectangle Y is three times taller than Rectangle X. The ratio of the lengths is 6:18, which means that the length of Rectangle Y is three times longer than Rectangle X.<\/p>\n<p>The scale factor is three because each side of Rectangle Y is three times that of Rectangle X.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat will happen to the surface area of the figure when the side lengths are increased by a scale factor of 4?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Proportional-Change-of-Dimensions-Example-2.svg\" alt=\"A rectangle labeled &quot;A&quot; with a length of 6 cm and a width of 2 cm.\" width=\"287.1\" height=\"98.1\" class=\"aligncenter size-full wp-image-287312\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">The surface area will increase by a scale factor of 12. <\/div><div class=\"PQ\"  id=\"PQ-2-2\">The surface area will increase by a scale factor of 10. <\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">The surface area will increase by a scale factor of 16. <\/div><div class=\"PQ\"  id=\"PQ-2-4\">The surface area will increase by a scale factor of 4. <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Squaring the scale factor of the side lengths will determine the scale factor of the areas. The side lengths of this shape are increased by a scale factor of 4; therefore, the area will be increased by a scale factor of \\(4^2\\), or 16.<\/p>\n<p>The new figure will have an area that is 16 times larger than the original figure. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat set of dimensions would create a rectangle that is \u201csimilar\u201d to the rectangle below?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Proportional-Change-of-Dimensions-Example-3.svg\" alt=\"A rectangle labeled with dimensions 30 feet wide and 60 feet long.\" width=\"196.35\" height=\"266.05\" class=\"aligncenter size-full wp-image-287306\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">Height: 50 ft<br>\r\nWidth: 20 ft <\/div><div class=\"PQ\"  id=\"PQ-3-2\">Height: 120 ft<br>\r\nWidth: 70 ft <\/div><div class=\"PQ\"  id=\"PQ-3-3\">Height: 15 ft<br>\r\nWidth: 20 ft <\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\">Height: 30 ft<br>\r\nWidth: 15 ft <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Shapes that are \u201csimilar\u201d have congruent corresponding angles, and proportional side lengths. In order to create side lengths that are proportional, we need to multiply both the height and the width by the same number.<\/p>\n<p>In this example, multiplying the height and the width by \\(\\frac{1}{2}\\) produces side lengths that are proportional.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMrs. Carson runs a concession stand at a movie theater. She has had many requests to create an extra large option for popcorn bags. Mrs. Carson agrees to start selling the jumbo size, and decides to double the dimensions of the current bag. The current bags are 7 inches tall, 4 inches wide, and 5 inches long. How will the volume of the new jumbo bags of popcorn compare to the smaller original bags? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">The new jumbo size popcorn bag will hold 8 times as much popcorn.<\/div><div class=\"PQ\"  id=\"PQ-4-2\">The new jumbo size popcorn bag will hold 9 times as much popcorn.<\/div><div class=\"PQ\"  id=\"PQ-4-3\">The new jumbo size popcorn bag will hold 10 times as much popcorn.<\/div><div class=\"PQ\"  id=\"PQ-4-4\">The new jumbo size popcorn bag will hold 11 times as much popcorn.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The new jumbo size bag dimensions increase by a scale factor of 2. Each side length is now twice as large as the original size. We can cube this scale factor in order to see what the scale factor in volume will be.<\/p>\n<p>A scale factor of 2 cubed gives us 8, because \\(2\\times2\\times2=8\\). This means that the new volume of popcorn will be 8 times as large as the original volume.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nCasey has a carton of juice that is 4 inches tall, 3 inches long, and 2 inches wide. If the dimensions were increased by a scale factor of 3, what would the new volume of juice be? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">648 in<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-5-2\">235 in<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-5-3\">455 in<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-5-4\">695 in<sup>3<\/sup><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The original dimensions of the carton of juice were multiplied by a scale factor of 3. If we cube this scale factor, we will know the scale factor for the volume.<\/p>\n<p style=\"text-align: center\">\\(3 \\times 3 \\times 3 = 27\\)<\/p>\n<p>so the carton of juice will be increased by a scale factor of 27. <\/p>\n<p>The original volume was \\(2 \\mathrm{\\:in} \\times 3 \\mathrm{\\:in} \\times 4 \\mathrm{\\:in}\\), which is 24 in<sup>3<\/sup>.<\/p>\n<p>When this volume is multiplied by a scale factor of 27, we have \\(24 \\mathrm{\\:in}^3  \\times 27 = 648 \\mathrm{\\:in}^3\\) as the new volume.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":100387,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-13252","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-proportion-videos","8":"page_category-ratios-and-proportions","9":"page_type-video","10":"content_type-practice-questions","11":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"Changing the size of 2D and 3D figures has different proportional effects on the shape. 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