{"id":4456,"date":"2020-04-13T13:41:44","date_gmt":"2020-04-13T13:41:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4456"},"modified":"2021-02-06T00:04:39","modified_gmt":"2021-02-06T00:04:39","slug":"finding-the-volume-and-surface-area-of-rectangular-solids","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/finding-the-volume-and-surface-area-of-rectangular-solids\/","title":{"raw":"Finding the Volume and Surface Area of Rectangular Solids","rendered":"Finding the Volume and Surface Area of Rectangular Solids"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the volume and surface area of a rectangular solid<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See the image below). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.<\/p>\r\nThis wooden crate is in the shape of a rectangular solid.\r\n\r\n<img class=\"aligncenter\" 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=\"This is an image of a wooden crate.\" \/>\r\nEach crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in the image below\u00a0has 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.\r\n\r\nBreaking 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.\r\n\r\n<img class=\"aligncenter\" 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 is shown. Each layer is 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 is an image of the top layer that says \" \/>\r\nAltogether 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]\r\n\r\n<img class=\"aligncenter\" 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.\" \/>\r\nThe volume, [latex]V[\/latex], of any rectangular solid is the product of the length, width, and height.\r\n\r\n[latex]V=LWH[\/latex]\r\n\r\nWe 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]\r\n\r\n[latex]B=L\\cdot W[\/latex]\r\n\r\nWe can substitute [latex]B[\/latex] for [latex]L\\cdot W[\/latex] in the volume formula to get another form of the volume formula.\r\n\r\n<img class=\"aligncenter\" 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.\" \/>\r\nWe 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.\r\n\r\n<img class=\"aligncenter\" 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=\"An image of a rectangular solid is shown. It is 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.\" \/>\r\nTo 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.\r\n\r\n[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]\r\n\r\nNotice for each of the three faces you see, there is an identical opposite face that does not show.\r\n\r\n[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]\r\n\r\nThe surface area [latex]S[\/latex] of the rectangular solid shown above\u00a0is [latex]52[\/latex] square units.\r\nIn 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.\r\n\r\n[latex]S=2LH+2LW+2WH[\/latex]\r\n\r\nFor each face of the rectangular solid facing you, there is another face on the opposite side. There are [latex]6[\/latex] faces in all.\r\n\r\n<img class=\"aligncenter\" 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 is shown. The sides are labeled L, W, and H. One face is labeled LW and another is labeled WH.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Volume and Surface Area of a Rectangular Solid<\/h3>\r\nFor a rectangular solid with length [latex]L[\/latex], width [latex]W[\/latex], and height [latex]H:[\/latex]\r\n\r\n<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 is shown. The sides are 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.\" \/>\r\n\r\n<\/div>\r\nDoing the Manipulative Mathematics activity \"Painted Cube\" will help you develop a better understanding of volume and surface area.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFor a rectangular solid with length [latex]14[\/latex] cm, height [latex]17[\/latex] cm, and width [latex]9[\/latex] cm, find the 1. volume and 2. surface area.\r\n\r\nSolution\r\nStep 1 is the same for both 1. and 2., so we will show it just once.\r\n<table id=\"eip-id1168468779989\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and\r\n\r\nlabel it with the given information.<\/td>\r\n<td><img 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=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468454551\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\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>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td>[latex]V=LWH[\/latex]\r\n\r\n[latex]V=\\mathrm{14}\\cdot 9\\cdot 17[\/latex]<\/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>Step 6. <strong>Check<\/strong>\r\n\r\nWe leave it to you to check your calculations.<\/td>\r\n<td><\/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]\\text{1,034}[\/latex] square centimeters.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469477419\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\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>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td>[latex]S=2LH+2LW+2WH[\/latex]\r\n\r\n[latex]S=2\\left(14\\cdot 17\\right)+2\\left(14\\cdot 9\\right)+2\\left(9\\cdot 17\\right)[\/latex]<\/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><\/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<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146790[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA rectangular crate has a length of [latex]30[\/latex] inches, width of [latex]25[\/latex] inches, and height of [latex]20[\/latex] inches. Find its 1. volume and 2. surface area.\r\n[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"57883\"]\r\n\r\nSolution\r\nStep 1 is the same for both 1. and 2., so we will show it just once.\r\n<table id=\"eip-id1168467275051\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and\r\n\r\nlabel it with the given information.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224141\/CNX_BMath_Figure_09_06_039_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467238054\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume of the crate<\/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>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td>[latex]V=LWH[\/latex]\r\n\r\n[latex]V=30\\cdot 25\\cdot 20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]V=15,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Double check your math.<\/td>\r\n<td><\/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]15,000[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467505864\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\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 crate<\/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>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td>[latex]S=2LH+2LW+2WH[\/latex]\r\n\r\n[latex]S=2\\left(30\\cdot 20\\right)+2\\left(30\\cdot 25\\right)+2\\left(25\\cdot 20\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]S=3,700[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Check it yourself!<\/td>\r\n<td><\/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]3,700[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146789[\/ohm_question]\r\n\r\n<\/div>\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. See Volume and Surface Area of a Cube, below. Substituting, <em>s<\/em> for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:<\/p>\r\n[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]\r\n\r\nSo for a cube, the formulas for volume and surface area are [latex]V={s}^{3}[\/latex] and [latex]S=6{s}^{2}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Volume and Surface Area of a Cube<\/h3>\r\nFor any cube with sides of length [latex]s[\/latex],\r\n\r\n<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=\"An image of a cube is shown. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA cube is [latex]2.5[\/latex] inches on each side. Find its 1. volume and 2. surface area.\r\n\r\n&nbsp;\r\n\r\nSolution\r\nStep 1 is the same for both 1. and 2., so we will show it just once.\r\n<table id=\"eip-id1168466154480\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and\r\n\r\nlabel it with the given information.<\/td>\r\n<td><img 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=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168465993815\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15.7656px\">\r\n<td style=\"height: 15.7656px\">1.<\/td>\r\n<td style=\"height: 15.7656px\"><\/td>\r\n<\/tr>\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\">Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.<\/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\">[latex]V={\\left(2.5\\right)}^{3}[\/latex]\r\n\r\n[latex]V=15.625[\/latex]<\/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\"><\/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<table id=\"eip-id1168466275896\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\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>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.<\/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>[latex]S=6\\cdot {\\left(2.5\\right)}^{2}[\/latex]\r\n\r\n[latex]S=37.5[\/latex]<\/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><\/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<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146791[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA notepad cube measures [latex]2[\/latex] inches on each side. Find its 1. volume and 2. surface area.\r\n[reveal-answer q=\"907715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"907715\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467296950\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and\r\n\r\nlabel it with the given information.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224144\/CNX_BMath_Figure_09_06_041_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467188434\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume 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>V<\/em> = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.<\/td>\r\n<td>[latex]V={s}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]V={2}^{3}[\/latex]\r\n\r\n[latex]V=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Check that you did the calculations\r\n\r\ncorrectly.<\/td>\r\n<td><\/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]8[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468478236\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\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>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.<\/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> the equation.<\/td>\r\n<td>[latex]S=6\\cdot {2}^{2}[\/latex]\r\n\r\n[latex]S=24[\/latex]<\/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><\/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]24[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146795[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the volume and surface area of a rectangular solid<\/li>\n<\/ul>\n<\/div>\n<p>A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See the image below). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.<\/p>\n<p>This wooden crate is in the shape of a rectangular solid.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" 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=\"This is an image of a wooden crate.\" \/><br \/>\nEach crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in the image below\u00a0has 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<p><img decoding=\"async\" class=\"aligncenter\" 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 is shown. Each layer is 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 is an image of the top layer that says\" \/><br \/>\nAltogether 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<p><img decoding=\"async\" class=\"aligncenter\" 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.\" \/><br \/>\nThe volume, [latex]V[\/latex], of any rectangular solid is the product of the length, width, and height.<\/p>\n<p>[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>[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<p><img decoding=\"async\" class=\"aligncenter\" 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.\" \/><br \/>\nWe 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<p><img decoding=\"async\" class=\"aligncenter\" 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=\"An image of a rectangular solid is shown. It is 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.\" \/><br \/>\nTo 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>[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>[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.<br \/>\nIn 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>[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<p><img decoding=\"async\" class=\"aligncenter\" 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 is shown. The sides are labeled L, W, and H. One face is labeled LW and another is labeled WH.\" \/><\/p>\n<div class=\"textbox shaded\">\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><img 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 is shown. The sides are 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.\" \/><\/p>\n<\/div>\n<p>Doing the Manipulative Mathematics activity &#8220;Painted Cube&#8221; will help you develop a better understanding of volume and surface area.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>For a rectangular solid with length [latex]14[\/latex] cm, height [latex]17[\/latex] cm, and width [latex]9[\/latex] cm, find the 1. volume and 2. surface area.<\/p>\n<p>Solution<br \/>\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>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/td>\n<td><img decoding=\"async\" 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=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468454551\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\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>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td>[latex]V=LWH[\/latex]<\/p>\n<p>[latex]V=\\mathrm{14}\\cdot 9\\cdot 17[\/latex]<\/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>Step 6. <strong>Check<\/strong><\/p>\n<p>We leave it to you to check your calculations.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is [latex]\\text{1,034}[\/latex] square centimeters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469477419\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\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>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td>[latex]S=2LH+2LW+2WH[\/latex]<\/p>\n<p>[latex]S=2\\left(14\\cdot 17\\right)+2\\left(14\\cdot 9\\right)+2\\left(9\\cdot 17\\right)[\/latex]<\/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><\/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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146790\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146790&theme=oea&iframe_resize_id=ohm146790&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A rectangular crate has a length of [latex]30[\/latex] inches, width of [latex]25[\/latex] inches, and height of [latex]20[\/latex] inches. Find its 1. volume and 2. surface area.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q57883\">Show Solution<\/span><\/p>\n<div id=\"q57883\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nStep 1 is the same for both 1. and 2., so we will show it just once.<\/p>\n<table id=\"eip-id1168467275051\" class=\"unnumbered unstyled\" summary=\"The text reads,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224141\/CNX_BMath_Figure_09_06_039_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467238054\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the crate<\/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>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td>[latex]V=LWH[\/latex]<\/p>\n<p>[latex]V=30\\cdot 25\\cdot 20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]V=15,000[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Double check your math.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is [latex]15,000[\/latex] cubic inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467505864\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the surface area of the crate<\/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>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td>[latex]S=2LH+2LW+2WH[\/latex]<\/p>\n<p>[latex]S=2\\left(30\\cdot 20\\right)+2\\left(30\\cdot 25\\right)+2\\left(25\\cdot 20\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]S=3,700[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Check it yourself!<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is [latex]3,700[\/latex] square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146789\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146789&theme=oea&iframe_resize_id=ohm146789&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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. See Volume and Surface Area of a Cube, below. Substituting, <em>s<\/em> for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:<\/p>\n<p>[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<div class=\"textbox shaded\">\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><img 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=\"An image of a cube is shown. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A cube is [latex]2.5[\/latex] inches on each side. Find its 1. volume and 2. surface area.<\/p>\n<p>&nbsp;<\/p>\n<p>Solution<br \/>\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>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/td>\n<td><img decoding=\"async\" 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=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168465993815\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr style=\"height: 15.7656px\">\n<td style=\"height: 15.7656px\">1.<\/td>\n<td style=\"height: 15.7656px\"><\/td>\n<\/tr>\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\">Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/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\">[latex]V={\\left(2.5\\right)}^{3}[\/latex]<\/p>\n<p>[latex]V=15.625[\/latex]<\/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\"><\/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<table id=\"eip-id1168466275896\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\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>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/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>[latex]S=6\\cdot {\\left(2.5\\right)}^{2}[\/latex]<\/p>\n<p>[latex]S=37.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> The check is left to you.<\/td>\n<td><\/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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146791\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146791&theme=oea&iframe_resize_id=ohm146791&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A notepad cube measures [latex]2[\/latex] inches on each side. Find its 1. volume and 2. surface area.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q907715\">Show Solution<\/span><\/p>\n<div id=\"q907715\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467296950\" class=\"unnumbered unstyled\" summary=\"The text reads,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224144\/CNX_BMath_Figure_09_06_041_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467188434\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume 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>V<\/em> = volume<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/td>\n<td>[latex]V={s}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]V={2}^{3}[\/latex]<\/p>\n<p>[latex]V=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Check that you did the calculations<\/p>\n<p>correctly.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is [latex]8[\/latex] cubic inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468478236\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\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>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/td>\n<td>[latex]S=6{s}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]S=6\\cdot {2}^{2}[\/latex]<\/p>\n<p>[latex]S=24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> The check is left to you.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is [latex]24[\/latex] square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146795\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146795&theme=oea&iframe_resize_id=ohm146795&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4456\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146790, 146789. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":25777,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146790, 146789\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4456","chapter","type-chapter","status-web-only","hentry"],"part":3706,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4456","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4456\/revisions"}],"predecessor-version":[{"id":5290,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4456\/revisions\/5290"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/3706"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4456\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4456"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4456"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4456"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}