{"id":1701,"date":"2024-04-24T16:58:22","date_gmt":"2024-04-24T16:58:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1701"},"modified":"2025-08-17T23:23:44","modified_gmt":"2025-08-17T23:23:44","slug":"analytical-applications-of-derivatives-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/analytical-applications-of-derivatives-background-youll-need-3\/","title":{"raw":"Analytical Applications of Derivatives: Background You'll Need 3","rendered":"Analytical Applications of Derivatives: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find the volume and surface area of different shapes &quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Find the volume and surface area of different shapes <\/span><\/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>\r\n[caption id=\"\" align=\"aligncenter\" width=\"439\"]<img 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\" \/> Rectangular solid with formulas for volume and surface area[\/caption]\r\n<center><\/center><\/center><\/div>\r\n<\/section>\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>\r\n[caption id=\"\" align=\"alignnone\" width=\"170\"]<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=\"A rectangular prism with one side labeled 14, one labeled 9, and another labeled 17\" width=\"170\" height=\"117\" \/> Rectangular prism with sides labeled[\/caption]\r\n<\/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\">[ohm_question hide_question_numbers=1]287789[\/ohm_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>\r\n[caption id=\"\" align=\"aligncenter\" width=\"272\"]<img 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\" \/> Cube with formulas for volume and surface area[\/caption]\r\n<\/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>\r\n[caption id=\"\" align=\"alignnone\" width=\"175\"]<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=\"A cube is shown with each side equal to 2.5\" width=\"175\" height=\"144\" \/> Cube with sides labeled[\/caption]\r\n<\/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\">[ohm_question hide_question_numbers=1]287790[\/ohm_question]<\/section>\r\n<section>\r\n<h2>Finding the Volume and Surface Area of a Sphere<\/h2>\r\n<p>A <strong>sphere<\/strong> is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below. Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>volume and surface area of a sphere<\/h3>\r\n<p>For a sphere with radius [latex]r\\text{:}[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"271\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224145\/CNX_BMath_Figure_09_06_015.png\" alt=\"A sphere, with radius labeled r. Beside this is Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.\" width=\"271\" height=\"94\" \/> Sphere with formulas for volume and surface area[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p><em>Note: We will approximate [latex]\\pi [\/latex] with [latex]3.14[\/latex].<\/em><\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">It is important to note that for both the volume and surface area of a sphere you use the radius of the sphere. Sometimes questions will only give you the diameter ([latex]d[\/latex]). To find the radius when given the diameter [latex]r = \\frac{d}{2}[\/latex]<\/section>\r\n<section class=\"textbox example\">A sphere has a radius [latex]6[\/latex] inches. Find its\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\r\n[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57883\"] Step 1 is the same for both 1. and 2., so we will show it just once.\r\n\r\n<table id=\"eip-id1168468504255\" 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 label it with the given information.<\/td>\r\n<td>\r\n[caption id=\"\" align=\"alignnone\" width=\"139\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224146\/CNX_BMath_Figure_09_06_042_img-01.png\" alt=\"A sphere with a radius of 6\" width=\"139\" height=\"132\" \/> Sphere with radius labeled[\/caption]\r\n<\/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 sphere<\/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> Write the appropriate formula.<\/td>\r\n<td>[latex]V=\\Large\\frac{4}{3}\\normalsize\\pi {r}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]V\\approx \\Large\\frac{4}{3}\\normalsize\\left(3.14\\right){6}^{3}[\/latex] [latex]V\\approx 904.32\\text{ cubic inches}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Double-check your math on 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 volume is approximately [latex]904.32[\/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>Step 4. <strong>Translate.<\/strong> Write the appropriate formula.<\/td>\r\n<td>[latex]S=4\\pi {r}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]S\\approx 4\\left(3.14\\right){6}^{2}[\/latex] [latex]S\\approx 452.16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Double-check your math on 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 approximately [latex]452.16[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]287791[\/ohm_question]<\/section>\r\n<section>\r\n<h2>Finding the Volume and Surface Area of a Cylinder<\/h2>\r\n<p>If you have ever seen a can of vegetables, you know what a cylinder looks like. A <strong>cylinder <\/strong>is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the <strong>bases<\/strong>. The <strong>height <\/strong>[latex]h[\/latex] of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, [latex]h[\/latex] , will be perpendicular to the bases.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"170\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224148\/CNX_BMath_Figure_09_06_020_img.png\" alt=\"A cylinder with an arrow pointing to the radius of the top labeling it r, radius. There is an arrow pointing to the height of the cylinder labeling it h, height.\" width=\"170\" height=\"156\" \/> Cylinder with radius and height labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>volume and surface area of a cylinder<\/h3>\r\n<p>For a cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex]:<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"329\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224155\/CNX_BMath_Figure_09_06_024.png\" alt=\"A cylinder, with the height labeled h and the radius of the top labeled r. Beside it is Volume: V equals pi times r squared times h or V equals capital B times h. Below this is Surface Area: S equals 2 times pi times r squared plus 2 times pi times r times h.\" width=\"329\" height=\"136\" \/> Cylinder with formulas for volume and surface area[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>For a cylinder, the area of the base, [latex]B[\/latex], is the area of its circular base, [latex]\\pi {r}^{2}[\/latex]. <em>This is different from the area of the base for rectangular solids.<\/em><\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">A cylinder has height [latex]5[\/latex] centimeters and radius [latex]3[\/latex] centimeters. Find its\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=\"57883\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57883\"]<br \/>\r\nStep 1 is the same for both 1. and 2., so we will show it just once.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label<\/p>\r\n<p>it with the given information.<\/p>\r\n<\/td>\r\n<td>\r\n[caption id=\"\" align=\"alignnone\" width=\"155\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224156\/CNX_BMath_Figure_09_06_046_img-01.png\" alt=\"A cylinder with height 5 and radius 3.\" width=\"155\" height=\"191\" \/> Cylinder with height and radius labeled[\/caption]\r\n<\/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 cylinder<\/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>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<p>Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] )<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]V=\\pi {r}^{2}h[\/latex]<\/p>\r\n<p>[latex]V\\approx \\left(3.14\\right){3}^{2}\\cdot 5[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]V\\approx 141.3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/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 approximately [latex]141.3[\/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 cylinder<\/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<p>Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] )<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]S=2\\pi {r}^{2}+2\\pi rh[\/latex]<\/p>\r\n<p>[latex]S\\approx 2\\left(3.14\\right){3}^{2}+2\\left(3.14\\right)\\left(3\\right)5[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]S\\approx 150.72[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/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 approximately [latex]150.72[\/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\">[ohm_question hide_question_numbers=1]287792[\/ohm_question]<\/section>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find the volume and surface area of different shapes &quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Find the volume and surface area of different shapes <\/span><\/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;\">\n<figure style=\"width: 439px\" 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\/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\" \/><figcaption class=\"wp-caption-text\">Rectangular solid with formulas for volume and surface area<\/figcaption><\/figure>\n<div style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/section>\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>\n<figure style=\"width: 170px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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=\"A rectangular prism with one side labeled 14, one labeled 9, and another labeled 17\" width=\"170\" height=\"117\" \/><figcaption class=\"wp-caption-text\">Rectangular prism with sides labeled<\/figcaption><\/figure>\n<\/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=\"ohm287789\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287789&theme=lumen&iframe_resize_id=ohm287789&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;\">\n<figure style=\"width: 272px\" 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\/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\" \/><figcaption class=\"wp-caption-text\">Cube with formulas for volume and surface area<\/figcaption><\/figure>\n<\/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>\n<figure style=\"width: 175px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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=\"A cube is shown with each side equal to 2.5\" width=\"175\" height=\"144\" \/><figcaption class=\"wp-caption-text\">Cube with sides labeled<\/figcaption><\/figure>\n<\/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=\"ohm287790\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287790&theme=lumen&iframe_resize_id=ohm287790&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Finding the Volume and Surface Area of a Sphere<\/h2>\n<p>A <strong>sphere<\/strong> is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below. Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>volume and surface area of a sphere<\/h3>\n<p>For a sphere with radius [latex]r\\text{:}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 271px\" 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\/24224145\/CNX_BMath_Figure_09_06_015.png\" alt=\"A sphere, with radius labeled r. Beside this is Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.\" width=\"271\" height=\"94\" \/><figcaption class=\"wp-caption-text\">Sphere with formulas for volume and surface area<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p><em>Note: We will approximate [latex]\\pi[\/latex] with [latex]3.14[\/latex].<\/em><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">It is important to note that for both the volume and surface area of a sphere you use the radius of the sphere. Sometimes questions will only give you the diameter ([latex]d[\/latex]). To find the radius when given the diameter [latex]r = \\frac{d}{2}[\/latex]<\/section>\n<section class=\"textbox example\">A sphere has a radius [latex]6[\/latex] inches. Find its<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>volume<\/li>\n<li>surface area<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57883\">Show Solution<\/button> <\/p>\n<div id=\"q57883\" class=\"hidden-answer\" style=\"display: none\"> Step 1 is the same for both 1. and 2., so we will show it just once.<\/p>\n<table id=\"eip-id1168468504255\" class=\"unnumbered unstyled\" summary=\"The text reads,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td>\n<figure style=\"width: 139px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224146\/CNX_BMath_Figure_09_06_042_img-01.png\" alt=\"A sphere with a radius of 6\" width=\"139\" height=\"132\" \/><figcaption class=\"wp-caption-text\">Sphere with radius labeled<\/figcaption><\/figure>\n<\/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 sphere<\/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> Write the appropriate formula.<\/td>\n<td>[latex]V=\\Large\\frac{4}{3}\\normalsize\\pi {r}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]V\\approx \\Large\\frac{4}{3}\\normalsize\\left(3.14\\right){6}^{3}[\/latex] [latex]V\\approx 904.32\\text{ cubic inches}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Double-check your math on a calculator.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is approximately [latex]904.32[\/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>Step 4. <strong>Translate.<\/strong> Write the appropriate formula.<\/td>\n<td>[latex]S=4\\pi {r}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]S\\approx 4\\left(3.14\\right){6}^{2}[\/latex] [latex]S\\approx 452.16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Double-check your math on 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 approximately [latex]452.16[\/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=\"ohm287791\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287791&theme=lumen&iframe_resize_id=ohm287791&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Finding the Volume and Surface Area of a Cylinder<\/h2>\n<p>If you have ever seen a can of vegetables, you know what a cylinder looks like. A <strong>cylinder <\/strong>is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the <strong>bases<\/strong>. The <strong>height <\/strong>[latex]h[\/latex] of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, [latex]h[\/latex] , will be perpendicular to the bases.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 170px\" 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\/24224148\/CNX_BMath_Figure_09_06_020_img.png\" alt=\"A cylinder with an arrow pointing to the radius of the top labeling it r, radius. There is an arrow pointing to the height of the cylinder labeling it h, height.\" width=\"170\" height=\"156\" \/><figcaption class=\"wp-caption-text\">Cylinder with radius and height labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>volume and surface area of a cylinder<\/h3>\n<p>For a cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex]:<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 329px\" 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\/24224155\/CNX_BMath_Figure_09_06_024.png\" alt=\"A cylinder, with the height labeled h and the radius of the top labeled r. Beside it is Volume: V equals pi times r squared times h or V equals capital B times h. Below this is Surface Area: S equals 2 times pi times r squared plus 2 times pi times r times h.\" width=\"329\" height=\"136\" \/><figcaption class=\"wp-caption-text\">Cylinder with formulas for volume and surface area<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>For a cylinder, the area of the base, [latex]B[\/latex], is the area of its circular base, [latex]\\pi {r}^{2}[\/latex]. <em>This is different from the area of the base for rectangular solids.<\/em><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">A cylinder has height [latex]5[\/latex] centimeters and radius [latex]3[\/latex] centimeters. Find its<\/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=\"q57883\">Show Solution<\/button><\/p>\n<div id=\"q57883\" 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>\n<tbody>\n<tr>\n<td>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label<\/p>\n<p>it with the given information.<\/p>\n<\/td>\n<td>\n<figure style=\"width: 155px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224156\/CNX_BMath_Figure_09_06_046_img-01.png\" alt=\"A cylinder with height 5 and radius 3.\" width=\"155\" height=\"191\" \/><figcaption class=\"wp-caption-text\">Cylinder with height and radius labeled<\/figcaption><\/figure>\n<\/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 cylinder<\/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>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi[\/latex] )<\/p>\n<\/td>\n<td>\n[latex]V=\\pi {r}^{2}h[\/latex]<br \/>\n[latex]V\\approx \\left(3.14\\right){3}^{2}\\cdot 5[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]V\\approx 141.3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is approximately [latex]141.3[\/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 cylinder<\/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<p>Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi[\/latex] )<\/p>\n<\/td>\n<td>\n[latex]S=2\\pi {r}^{2}+2\\pi rh[\/latex]<br \/>\n[latex]S\\approx 2\\left(3.14\\right){3}^{2}+2\\left(3.14\\right)\\left(3\\right)5[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]S\\approx 150.72[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/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 approximately [latex]150.72[\/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=\"ohm287792\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287792&theme=lumen&iframe_resize_id=ohm287792&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/section>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"module-header":"background_you_need","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1701"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1701\/revisions"}],"predecessor-version":[{"id":4772,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1701\/revisions\/4772"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1701\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1701"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1701"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1701"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}