{"id":2765,"date":"2023-05-15T17:35:16","date_gmt":"2023-05-15T17:35:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2765"},"modified":"2025-08-26T03:52:57","modified_gmt":"2025-08-26T03:52:57","slug":"volume-and-surface-area-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/volume-and-surface-area-learn-it-1\/","title":{"raw":"Volume and Surface Area: Learn It 1","rendered":"Volume and Surface Area: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find the volume of a rectangular solid, sphere, cylinder, and cone<\/li>\r\n\t<li>Find the surface area of a rectangular solid, sphere, and cylinder<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Finding the Volume and Surface Area of Rectangular Solids<\/h2>\r\n<p>When we explore three-dimensional shapes, understanding how to calculate the volume and surface area is crucial. Volume measures the space a shape occupies, while surface area describes the total area of all the surfaces of a three-dimensional object. For rectangular solids, which include cubes and rectangular prisms, these measurements are based on the object's length, width, and height.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>volume and surface area of a rectangular solid<\/h3>\r\n<p>For a rectangular solid with length [latex]L[\/latex], width [latex]W[\/latex], and height [latex]H[\/latex]:<\/p>\r\n<p>&nbsp;<\/p>\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224139\/CNX_BMath_Figure_09_06_006_img.png\" alt=\"A rectangular solid, with sides labeled L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.\" width=\"439\" height=\"174\" \/><center><\/center><\/center><\/div>\r\n<\/section>\r\n<p>How did we come up with the two equations above? Let's break down a problem to find out.<\/p>\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"374\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224130\/CNX_BMath_Figure_09_06_001.png\" alt=\"A painted wooden crate\" width=\"374\" height=\"227\" \/> Figure 1. The amount of space within the crate is the volume[\/caption]\r\n\r\n<p><br \/>\r\nA cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. The amount of paint needed to cover the outside of each box is the <strong>surface area<\/strong>, a square measure of the total area of all the sides. The amount of space inside the crate is the <strong>volume<\/strong>, a cubic measure.<\/p>\r\n<p>This wooden crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in the image to the right has length [latex]4[\/latex] units, width [latex]2[\/latex] units, and height [latex]3[\/latex] units. Can you tell how many cubic units there are altogether? Let\u2019s look layer by layer.<\/p>\r\n<p>Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This [latex]4[\/latex] by [latex]2[\/latex] by [latex]3[\/latex] rectangular solid has [latex]24[\/latex] cubic units.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"507\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224132\/CNX_BMath_Figure_09_06_002.png\" alt=\"A rectangular solid, with each layer composed of 8 cubes, measuring 2 by 4. The top layer is pink. The middle layer is orange. The bottom layer is green. Beside this, each layer is separated and there is text reading 'the top layer has 8 cubic units, the middle layer has 8 cubic units, and the bottom layer has 8 cubic units.'\" width=\"507\" height=\"133\" \/> Figure 2. The volume is measured in cubic units[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Altogether there are [latex]24[\/latex] cubic units. Notice that [latex]24[\/latex] is the [latex]\\text{length}\\times \\text{width}\\times \\text{height}\\text{.}[\/latex]<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"265\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224134\/CNX_BMath_Figure_09_06_002_img.png\" alt=\"The top line says V equals L times W times H. Beneath the V is 24, beneath the equal sign is another equal sign, beneath the L is a 4, beneath the W is a 2, beneath the H is a 3.\" width=\"265\" height=\"57\" \/> Figure 3. The volume, V, is equal to the length times the width times the height[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The volume, [latex]V[\/latex], of any rectangular solid is the product of the length, width, and height.<\/p>\r\n<p style=\"text-align: center;\">[latex]V=LWH[\/latex]<\/p>\r\n<p>We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, [latex]B[\/latex], is equal to [latex]\\text{length}\\times \\text{width}\\text{.}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]B=L\\cdot W[\/latex]<\/p>\r\n<p>We can substitute [latex]B[\/latex] for [latex]L\\cdot W[\/latex] in the volume formula to get another form of the volume formula.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"104\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224135\/CNX_BMath_Figure_09_06_003_img.png\" alt=\"The top line says V equals red L times red W times H. Below this is V equals red parentheses L times W times H. Below this is V equals red capital B times h.\" width=\"104\" height=\"72\" \/> Figure 4. The volume's formula is also equal to the area of the base, B, times the height, h[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>We now have another version of the volume formula for rectangular solids. Let\u2019s see how this works with the [latex]4\\times 2\\times 3[\/latex] rectangular solid we started with. See the image below.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"302\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224136\/CNX_BMath_Figure_09_06_004_img.png\" alt=\"A rectangular solid made up of cubes. It is labeled as 2 by 4 by 3. Beside the solid is V equals Bh. Below this is V equals Base times height. Below Base is parentheses 4 times 2. The next line says V equals parentheses 4 times 2 times 3. Below that is V equals 8 times 3, then V equals 24 cubic units.\" width=\"302\" height=\"118\" \/> Figure 5. The volume is 24 cubic units[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>To find the <em>surface area<\/em> of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccccc}{A}_{\\text{front}}=L\\times W\\hfill &amp; &amp; &amp; {A}_{\\text{side}}=L\\times W\\hfill &amp; &amp; &amp; {A}_{\\text{top}}=L\\times W\\hfill \\\\ {A}_{\\text{front}}=4\\cdot 3\\hfill &amp; &amp; &amp; {A}_{\\text{side}}=2\\cdot 3\\hfill &amp; &amp; &amp; {A}_{\\text{top}}=4\\cdot 2\\hfill \\\\ {A}_{\\text{front}}=12\\hfill &amp; &amp; &amp; {A}_{\\text{side}}=6\\hfill &amp; &amp; &amp; {A}_{\\text{top}}=8\\hfill \\end{array}[\/latex]<\/p>\r\n<p>Notice for each of the three faces you see, there is an identical opposite face that does not show.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}S=\\left(\\text{front}+\\text{back}\\right)\\text{+}\\left(\\text{left side}+\\text{right side}\\right)+\\left(\\text{top}+\\text{bottom}\\right)\\\\ S=\\left(2\\cdot \\text{front}\\right)+\\left(\\text{2}\\cdot \\text{left side}\\right)+\\left(\\text{2}\\cdot \\text{top}\\right)\\\\ S=2\\cdot 12+2\\cdot 6+2\\cdot 8\\\\ S=24+12+16\\\\ S=52\\text{ sq. units}\\end{array}[\/latex]<\/p>\r\n<p>The surface area [latex]S[\/latex] of the rectangular solid shown above\u00a0is [latex]52[\/latex] square units.<\/p>\r\n<p>In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see the image below). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.<\/p>\r\n<p style=\"text-align: center;\">[latex]S=2LH+2LW+2WH[\/latex]<\/p>\r\n<p>For each face of the rectangular solid facing you, there is another face on the opposite side. There are [latex]6[\/latex] faces in all.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"156\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224137\/CNX_BMath_Figure_09_06_005.png\" alt=\"A rectangular solid, with sides labeled L, W, and H. One face is labeled LW and another is labeled WH.\" width=\"156\" height=\"173\" \/> Figure 6. The rectangular solid is labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox example\">For a rectangular solid with length [latex]14[\/latex] cm, height [latex]17[\/latex] cm, and width [latex]9[\/latex] cm. Find the:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>volume<\/li>\r\n\t<li>surface area<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"4331\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4331\"]<br \/>\r\nStep 1 is the same for both 1. and 2., so we will show it just once.<\/p>\r\n<table id=\"eip-id1168468779989\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\r\n<p>label it with the given information.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224140\/CNX_BMath_Figure_09_06_038_img-01.png\" alt=\"A rectangular prism with one side labeled 14, one labeled 9, and another labeled 17\" width=\"170\" height=\"117\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume of the rectangular solid<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]V[\/latex] = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<p>Substitute.<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]V=LWH[\/latex]<\/p>\r\n<p>[latex]V=\\mathrm{14}\\cdot 9\\cdot 17[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]V=2,142[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 6. <strong>Check<\/strong><\/p>\r\n<p>We leave it to you to check your calculations.<\/p>\r\n<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The volume is [latex]2,142[\/latex] cubic centimeters.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the surface area of the solid<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]S[\/latex] = surface area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<p>Substitute.<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]S=2LH+2LW+2WH[\/latex]<\/p>\r\n<p>[latex]S=2\\left(14\\cdot 17\\right)+2\\left(14\\cdot 9\\right)+2\\left(9\\cdot 17\\right)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve the equation.<\/strong><\/td>\r\n<td>[latex]S=1,034[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Double-check with a calculator.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The surface area is [latex]1,034[\/latex] square centimeters.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6968[\/ohm2_question]<\/section>\r\n<h2>Finding the Volume and Surface Area of a Cube<\/h2>\r\n<p>A cube is a rectangular solid whose length, width, and height are equal. Substituting, [latex]s[\/latex] for the length, width, and height into the formulas for volume and surface area of a rectangular solid, we get:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}V=LWH\\hfill &amp; &amp; &amp; &amp; S=2LH+2LW+2WH\\hfill \\\\ V=s\\cdot s\\cdot s\\hfill &amp; &amp; &amp; &amp; S=2s\\cdot s+2s\\cdot s+2s\\cdot s\\hfill \\\\ V={s}^{3}\\hfill &amp; &amp; &amp; &amp; S=2{s}^{2}+2{s}^{2}+2{s}^{2}\\hfill \\\\ &amp; &amp; &amp; &amp; S=6{s}^{2}\\hfill \\end{array}[\/latex]<\/p>\r\n<p>So for a cube, the formulas for volume and surface area are [latex]V={s}^{3}[\/latex] and [latex]S=6{s}^{2}[\/latex].<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>volume and surface area of a cube<\/h3>\r\n<p>For any cube with sides of length [latex]s[\/latex]:<\/p>\r\n<p>&nbsp;<\/p>\r\n<br \/>\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224142\/CNX_BMath_Figure_09_06_010_img.png\" alt=\"A cube. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.\" width=\"272\" height=\"104\" \/><\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox example\">A cube is [latex]2.5[\/latex] inches on each side. Find the:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>volume<\/li>\r\n\t<li>surface area<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"4330\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4330\"]<br \/>\r\nStep 1 is the same for both 1. and 2., so we will show it just once.<\/p>\r\n<table id=\"eip-id1168466154480\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\r\n<p>label it with the given information.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224143\/CNX_BMath_Figure_09_06_040_img-01.png\" alt=\"A cube is shown with each side equal to 2.5\" width=\"175\" height=\"144\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td style=\"height: 15px;\">the volume of the cube<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td style=\"height: 15px;\">let <em>V<\/em> = volume<\/td>\r\n<\/tr>\r\n<tr style=\"height: 59px;\">\r\n<td style=\"height: 59px;\">\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<\/td>\r\n<td style=\"height: 59px;\">[latex]V={s}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 58px;\">\r\n<td style=\"height: 58px;\">Step 5. <strong>Solve.<\/strong> Substitute and solve.<\/td>\r\n<td style=\"height: 58px;\">\r\n<p>[latex]V={\\left(2.5\\right)}^{3}[\/latex]<\/p>\r\n<p>[latex]V=15.625[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Step 6. <strong>Check:<\/strong> Check your work.<\/td>\r\n<td style=\"height: 15px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td style=\"height: 15px;\">The volume is [latex]15.625[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the surface area of the cube<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>S<\/em> = surface area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<\/td>\r\n<td>[latex]S=6{s}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong> Substitute and solve.<\/td>\r\n<td>\r\n<p>[latex]S=6\\cdot {\\left(2.5\\right)}^{2}[\/latex]<\/p>\r\n<p>[latex]S=37.5[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> The check is left to you.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The surface area is [latex]37.5[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6982[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the volume of a rectangular solid, sphere, cylinder, and cone<\/li>\n<li>Find the surface area of a rectangular solid, sphere, and cylinder<\/li>\n<\/ul>\n<\/section>\n<h2>Finding the Volume and Surface Area of Rectangular Solids<\/h2>\n<p>When we explore three-dimensional shapes, understanding how to calculate the volume and surface area is crucial. Volume measures the space a shape occupies, while surface area describes the total area of all the surfaces of a three-dimensional object. For rectangular solids, which include cubes and rectangular prisms, these measurements are based on the object&#8217;s length, width, and height.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>volume and surface area of a rectangular solid<\/h3>\n<p>For a rectangular solid with length [latex]L[\/latex], width [latex]W[\/latex], and height [latex]H[\/latex]:<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224139\/CNX_BMath_Figure_09_06_006_img.png\" alt=\"A rectangular solid, with sides labeled L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.\" width=\"439\" height=\"174\" \/><\/p>\n<div style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>How did we come up with the two equations above? Let&#8217;s break down a problem to find out.<\/p>\n<figure style=\"width: 374px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224130\/CNX_BMath_Figure_09_06_001.png\" alt=\"A painted wooden crate\" width=\"374\" height=\"227\" \/><figcaption class=\"wp-caption-text\">Figure 1. The amount of space within the crate is the volume<\/figcaption><\/figure>\n<p>\nA cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. The amount of paint needed to cover the outside of each box is the <strong>surface area<\/strong>, a square measure of the total area of all the sides. The amount of space inside the crate is the <strong>volume<\/strong>, a cubic measure.<\/p>\n<p>This wooden crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in the image to the right has length [latex]4[\/latex] units, width [latex]2[\/latex] units, and height [latex]3[\/latex] units. Can you tell how many cubic units there are altogether? Let\u2019s look layer by layer.<\/p>\n<p>Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This [latex]4[\/latex] by [latex]2[\/latex] by [latex]3[\/latex] rectangular solid has [latex]24[\/latex] cubic units.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 507px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224132\/CNX_BMath_Figure_09_06_002.png\" alt=\"A rectangular solid, with each layer composed of 8 cubes, measuring 2 by 4. The top layer is pink. The middle layer is orange. The bottom layer is green. Beside this, each layer is separated and there is text reading 'the top layer has 8 cubic units, the middle layer has 8 cubic units, and the bottom layer has 8 cubic units.'\" width=\"507\" height=\"133\" \/><figcaption class=\"wp-caption-text\">Figure 2. The volume is measured in cubic units<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Altogether there are [latex]24[\/latex] cubic units. Notice that [latex]24[\/latex] is the [latex]\\text{length}\\times \\text{width}\\times \\text{height}\\text{.}[\/latex]<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 265px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224134\/CNX_BMath_Figure_09_06_002_img.png\" alt=\"The top line says V equals L times W times H. Beneath the V is 24, beneath the equal sign is another equal sign, beneath the L is a 4, beneath the W is a 2, beneath the H is a 3.\" width=\"265\" height=\"57\" \/><figcaption class=\"wp-caption-text\">Figure 3. The volume, V, is equal to the length times the width times the height<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The volume, [latex]V[\/latex], of any rectangular solid is the product of the length, width, and height.<\/p>\n<p style=\"text-align: center;\">[latex]V=LWH[\/latex]<\/p>\n<p>We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, [latex]B[\/latex], is equal to [latex]\\text{length}\\times \\text{width}\\text{.}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]B=L\\cdot W[\/latex]<\/p>\n<p>We can substitute [latex]B[\/latex] for [latex]L\\cdot W[\/latex] in the volume formula to get another form of the volume formula.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 104px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224135\/CNX_BMath_Figure_09_06_003_img.png\" alt=\"The top line says V equals red L times red W times H. Below this is V equals red parentheses L times W times H. Below this is V equals red capital B times h.\" width=\"104\" height=\"72\" \/><figcaption class=\"wp-caption-text\">Figure 4. The volume&#8217;s formula is also equal to the area of the base, B, times the height, h<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>We now have another version of the volume formula for rectangular solids. Let\u2019s see how this works with the [latex]4\\times 2\\times 3[\/latex] rectangular solid we started with. See the image below.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 302px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224136\/CNX_BMath_Figure_09_06_004_img.png\" alt=\"A rectangular solid made up of cubes. It is labeled as 2 by 4 by 3. Beside the solid is V equals Bh. Below this is V equals Base times height. Below Base is parentheses 4 times 2. The next line says V equals parentheses 4 times 2 times 3. Below that is V equals 8 times 3, then V equals 24 cubic units.\" width=\"302\" height=\"118\" \/><figcaption class=\"wp-caption-text\">Figure 5. The volume is 24 cubic units<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To find the <em>surface area<\/em> of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccccc}{A}_{\\text{front}}=L\\times W\\hfill & & & {A}_{\\text{side}}=L\\times W\\hfill & & & {A}_{\\text{top}}=L\\times W\\hfill \\\\ {A}_{\\text{front}}=4\\cdot 3\\hfill & & & {A}_{\\text{side}}=2\\cdot 3\\hfill & & & {A}_{\\text{top}}=4\\cdot 2\\hfill \\\\ {A}_{\\text{front}}=12\\hfill & & & {A}_{\\text{side}}=6\\hfill & & & {A}_{\\text{top}}=8\\hfill \\end{array}[\/latex]<\/p>\n<p>Notice for each of the three faces you see, there is an identical opposite face that does not show.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}S=\\left(\\text{front}+\\text{back}\\right)\\text{+}\\left(\\text{left side}+\\text{right side}\\right)+\\left(\\text{top}+\\text{bottom}\\right)\\\\ S=\\left(2\\cdot \\text{front}\\right)+\\left(\\text{2}\\cdot \\text{left side}\\right)+\\left(\\text{2}\\cdot \\text{top}\\right)\\\\ S=2\\cdot 12+2\\cdot 6+2\\cdot 8\\\\ S=24+12+16\\\\ S=52\\text{ sq. units}\\end{array}[\/latex]<\/p>\n<p>The surface area [latex]S[\/latex] of the rectangular solid shown above\u00a0is [latex]52[\/latex] square units.<\/p>\n<p>In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see the image below). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.<\/p>\n<p style=\"text-align: center;\">[latex]S=2LH+2LW+2WH[\/latex]<\/p>\n<p>For each face of the rectangular solid facing you, there is another face on the opposite side. There are [latex]6[\/latex] faces in all.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 156px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224137\/CNX_BMath_Figure_09_06_005.png\" alt=\"A rectangular solid, with sides labeled L, W, and H. One face is labeled LW and another is labeled WH.\" width=\"156\" height=\"173\" \/><figcaption class=\"wp-caption-text\">Figure 6. The rectangular solid is labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox example\">For a rectangular solid with length [latex]14[\/latex] cm, height [latex]17[\/latex] cm, and width [latex]9[\/latex] cm. Find the:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>volume<\/li>\n<li>surface area<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4331\">Show Solution<\/button><\/p>\n<div id=\"q4331\" class=\"hidden-answer\" style=\"display: none\">\nStep 1 is the same for both 1. and 2., so we will show it just once.<\/p>\n<table id=\"eip-id1168468779989\" class=\"unnumbered unstyled\" summary=\"The text reads,\">\n<tbody>\n<tr>\n<td>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224140\/CNX_BMath_Figure_09_06_038_img-01.png\" alt=\"A rectangular prism with one side labeled 14, one labeled 9, and another labeled 17\" width=\"170\" height=\"117\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: decimal;\">\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the rectangular solid<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]V[\/latex] = volume<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/p>\n<\/td>\n<td>\n[latex]V=LWH[\/latex]<br \/>\n[latex]V=\\mathrm{14}\\cdot 9\\cdot 17[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]V=2,142[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 6. <strong>Check<\/strong><\/p>\n<p>We leave it to you to check your calculations.<\/p>\n<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is [latex]2,142[\/latex] cubic centimeters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the surface area of the solid<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]S[\/latex] = surface area<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/p>\n<\/td>\n<td>\n[latex]S=2LH+2LW+2WH[\/latex]<br \/>\n[latex]S=2\\left(14\\cdot 17\\right)+2\\left(14\\cdot 9\\right)+2\\left(9\\cdot 17\\right)[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve the equation.<\/strong><\/td>\n<td>[latex]S=1,034[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Double-check with a calculator.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is [latex]1,034[\/latex] square centimeters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6968\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6968&theme=lumen&iframe_resize_id=ohm6968&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Finding the Volume and Surface Area of a Cube<\/h2>\n<p>A cube is a rectangular solid whose length, width, and height are equal. Substituting, [latex]s[\/latex] for the length, width, and height into the formulas for volume and surface area of a rectangular solid, we get:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}V=LWH\\hfill & & & & S=2LH+2LW+2WH\\hfill \\\\ V=s\\cdot s\\cdot s\\hfill & & & & S=2s\\cdot s+2s\\cdot s+2s\\cdot s\\hfill \\\\ V={s}^{3}\\hfill & & & & S=2{s}^{2}+2{s}^{2}+2{s}^{2}\\hfill \\\\ & & & & S=6{s}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>So for a cube, the formulas for volume and surface area are [latex]V={s}^{3}[\/latex] and [latex]S=6{s}^{2}[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>volume and surface area of a cube<\/h3>\n<p>For any cube with sides of length [latex]s[\/latex]:<\/p>\n<p>&nbsp;<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224142\/CNX_BMath_Figure_09_06_010_img.png\" alt=\"A cube. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.\" width=\"272\" height=\"104\" \/><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">A cube is [latex]2.5[\/latex] inches on each side. Find the:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>volume<\/li>\n<li>surface area<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4330\">Show Solution<\/button><\/p>\n<div id=\"q4330\" class=\"hidden-answer\" style=\"display: none\">\nStep 1 is the same for both 1. and 2., so we will show it just once.<\/p>\n<table id=\"eip-id1168466154480\" class=\"unnumbered unstyled\" summary=\"The text reads,\">\n<tbody>\n<tr>\n<td>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224143\/CNX_BMath_Figure_09_06_040_img-01.png\" alt=\"A cube is shown with each side equal to 2.5\" width=\"175\" height=\"144\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: decimal;\">\n<li>\n<table>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td style=\"height: 15px;\">the volume of the cube<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td style=\"height: 15px;\">let <em>V<\/em> = volume<\/td>\n<\/tr>\n<tr style=\"height: 59px;\">\n<td style=\"height: 59px;\">\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<\/td>\n<td style=\"height: 59px;\">[latex]V={s}^{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 58px;\">\n<td style=\"height: 58px;\">Step 5. <strong>Solve.<\/strong> Substitute and solve.<\/td>\n<td style=\"height: 58px;\">\n[latex]V={\\left(2.5\\right)}^{3}[\/latex]<br \/>\n[latex]V=15.625[\/latex]\n<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Step 6. <strong>Check:<\/strong> Check your work.<\/td>\n<td style=\"height: 15px;\">\u00a0<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td style=\"height: 15px;\">The volume is [latex]15.625[\/latex] cubic inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the surface area of the cube<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>S<\/em> = surface area<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<\/td>\n<td>[latex]S=6{s}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong> Substitute and solve.<\/td>\n<td>\n[latex]S=6\\cdot {\\left(2.5\\right)}^{2}[\/latex]<br \/>\n[latex]S=37.5[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> The check is left to you.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is [latex]37.5[\/latex] square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6982\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6982&theme=lumen&iframe_resize_id=ohm6982&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2765"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":56,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2765\/revisions"}],"predecessor-version":[{"id":15658,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2765\/revisions\/15658"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/71"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2765\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2765"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2765"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2765"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}