{"id":10782,"date":"2017-06-05T16:18:17","date_gmt":"2017-06-05T16:18:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10782"},"modified":"2018-01-13T19:37:22","modified_gmt":"2018-01-13T19:37:22","slug":"finding-the-volume-of-a-cone","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/finding-the-volume-of-a-cone\/","title":{"raw":"Finding the Volume of a Cone","rendered":"Finding the Volume of a Cone"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the volume of a cone<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p data-type=\"title\">The first image that many of us have when we hear the word \u2018cone\u2019 is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.<\/p>\r\n<p data-type=\"title\">In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See the image below.<\/p>\r\nThe height of a cone is the distance between its base and the vertex.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224158\/CNX_BMath_Figure_09_06_029.png\" alt=\"An image of a cone is shown. The top is labeled vertex. The height is labeled h. The radius of the base is labeled r.\" data-media-type=\"image\/png\" \/>\r\nEarlier in this section, we saw that the volume of a cylinder is [latex]V=\\text{\\pi }{r}^{2}h[\/latex]. We can think of a cone as part of a cylinder. The image below shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.\r\n\r\nThe volume of a cone is less than the volume of a cylinder with the same base and height.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224159\/CNX_BMath_Figure_09_06_030.png\" alt=\"An image of a cone is shown. There is a cylinder drawn around it.\" data-media-type=\"image\/png\" \/>\r\nIn fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224200\/CNX_BMath_Figure_09_06_031_img.png\" alt=\"The formula V equals one-third times capital B times h is shown.\" data-media-type=\"image\/png\" \/>\r\nSince the base of a cone is a circle, we can substitute the formula of area of a circle, [latex]\\text{\\pi }{r}^{2}[\/latex] , for [latex]B[\/latex] to get the formula for volume of a cone.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224200\/CNX_BMath_Figure_09_06_032_img.png\" alt=\"The formula V equals one-third times pi times r squared times h is shown.\" data-media-type=\"image\/png\" \/>\r\nIn this book, we will only find the volume of a cone, and not its surface area.\r\n<div class=\"textbox shaded\">\r\n<h3>Volume of a Cone<\/h3>\r\nFor a cone with radius [latex]r[\/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\/24224201\/CNX_BMath_Figure_09_06_033_img.png\" alt=\"An image of a cone is shown. The height is labeled h, the radius of the base is labeled r. Beside this is Volume: V equals one-third times pi times r squared times h.\" data-media-type=\"image\/png\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the volume of a cone with height [latex]6[\/latex] inches and radius of its base [latex]2[\/latex] inches.\r\n\r\nSolution\r\n<table id=\"eip-id1168467046430\" class=\"unnumbered unstyled\" summary=\"The text reads, \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it\r\n\r\nwith 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\/24224203\/CNX_BMath_Figure_09_06_048_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/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 cone<\/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. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] )<\/td>\r\n<td>[latex]V=\\Large\\frac{1}{3}\\normalsize\\pi {r}^{2}h[\/latex]\r\n\r\n[latex]V\\approx \\Large\\frac{1}{3}\\normalsize 3.14{\\left(2\\right)}^{2}\\left(6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]V\\approx 25.12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your\r\n\r\ncalculations.<\/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 approximately [latex]25.12[\/latex] cubic 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]146818[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMarty\u2019s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is [latex]8[\/latex] inches tall and [latex]5[\/latex] inches in diameter? Round the answer to the nearest hundredth.\r\n\r\n[reveal-answer q=\"499965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"499965\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469874277\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \" data-label=\"\">\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. Notice here that the base is the circle at the top of the cone.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224204\/CNX_BMath_Figure_09_06_049_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/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 cone<\/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 data-effect=\"italics\">V<\/em> = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] , and notice that we were given the distance across the circle, which is its diameter. The radius is [latex]2.5[\/latex] inches.)<\/td>\r\n<td>[latex]V=\\Large\\frac{1}{3}\\normalsize\\pi {r}^{2}h[\/latex]\r\n\r\n[latex]V\\approx \\Large\\frac{1}{3}\\normalsize 3.14{\\left(2.5\\right)}^{2}\\left(8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]V\\approx 52.33[\/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><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The volume of the wrap is approximately [latex]52.33[\/latex] cubic 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]146820[\/ohm_question]\r\n\r\n<\/div>\r\n<p data-type=\"title\">\u00a0In the following video we provide another example of how to find the volume of a cone.<\/p>\r\nhttps:\/\/youtu.be\/7Y0ZMnCcVGs","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the volume of a cone<\/li>\n<\/ul>\n<\/div>\n<p data-type=\"title\">The first image that many of us have when we hear the word \u2018cone\u2019 is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.<\/p>\n<p data-type=\"title\">In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See the image below.<\/p>\n<p>The height of a cone is the distance between its base and the vertex.<\/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\/24224158\/CNX_BMath_Figure_09_06_029.png\" alt=\"An image of a cone is shown. The top is labeled vertex. The height is labeled h. The radius of the base is labeled r.\" data-media-type=\"image\/png\" \/><br \/>\nEarlier in this section, we saw that the volume of a cylinder is [latex]V=\\text{\\pi }{r}^{2}h[\/latex]. We can think of a cone as part of a cylinder. The image below shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.<\/p>\n<p>The volume of a cone is less than the volume of a cylinder with the same base and height.<\/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\/24224159\/CNX_BMath_Figure_09_06_030.png\" alt=\"An image of a cone is shown. There is a cylinder drawn around it.\" data-media-type=\"image\/png\" \/><br \/>\nIn fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is<\/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\/24224200\/CNX_BMath_Figure_09_06_031_img.png\" alt=\"The formula V equals one-third times capital B times h is shown.\" data-media-type=\"image\/png\" \/><br \/>\nSince the base of a cone is a circle, we can substitute the formula of area of a circle, [latex]\\text{\\pi }{r}^{2}[\/latex] , for [latex]B[\/latex] to get the formula for volume of a cone.<\/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\/24224200\/CNX_BMath_Figure_09_06_032_img.png\" alt=\"The formula V equals one-third times pi times r squared times h is shown.\" data-media-type=\"image\/png\" \/><br \/>\nIn this book, we will only find the volume of a cone, and not its surface area.<\/p>\n<div class=\"textbox shaded\">\n<h3>Volume of a Cone<\/h3>\n<p>For a cone with radius [latex]r[\/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\/24224201\/CNX_BMath_Figure_09_06_033_img.png\" alt=\"An image of a cone is shown. The height is labeled h, the radius of the base is labeled r. Beside this is Volume: V equals one-third times pi times r squared times h.\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the volume of a cone with height [latex]6[\/latex] inches and radius of its base [latex]2[\/latex] inches.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467046430\" class=\"unnumbered unstyled\" summary=\"The text reads,\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it<\/p>\n<p>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\/24224203\/CNX_BMath_Figure_09_06_048_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the cone<\/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. (Use [latex]3.14[\/latex] for [latex]\\pi[\/latex] )<\/td>\n<td>[latex]V=\\Large\\frac{1}{3}\\normalsize\\pi {r}^{2}h[\/latex]<\/p>\n<p>[latex]V\\approx \\Large\\frac{1}{3}\\normalsize 3.14{\\left(2\\right)}^{2}\\left(6\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]V\\approx 25.12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your<\/p>\n<p>calculations.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is approximately [latex]25.12[\/latex] cubic 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=\"ohm146818\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146818&theme=oea&iframe_resize_id=ohm146818&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>Marty\u2019s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is [latex]8[\/latex] inches tall and [latex]5[\/latex] inches in diameter? Round the answer to the nearest hundredth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q499965\">Show Solution<\/span><\/p>\n<div id=\"q499965\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469874277\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224204\/CNX_BMath_Figure_09_06_049_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the cone<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em data-effect=\"italics\">V<\/em> = volume<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi[\/latex] , and notice that we were given the distance across the circle, which is its diameter. The radius is [latex]2.5[\/latex] inches.)<\/td>\n<td>[latex]V=\\Large\\frac{1}{3}\\normalsize\\pi {r}^{2}h[\/latex]<\/p>\n<p>[latex]V\\approx \\Large\\frac{1}{3}\\normalsize 3.14{\\left(2.5\\right)}^{2}\\left(8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]V\\approx 52.33[\/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><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume of the wrap is approximately [latex]52.33[\/latex] cubic 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=\"ohm146820\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146820&theme=oea&iframe_resize_id=ohm146820&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p data-type=\"title\">\u00a0In the following video we provide another example of how to find the volume of a cone.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Determine Volume of a Cone\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7Y0ZMnCcVGs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-10782\">\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 146820, 146818. <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, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Determine Volume of a Cone. <strong>Authored by<\/strong>: James Sousa (amthispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/7Y0ZMnCcVGs\">https:\/\/youtu.be\/7Y0ZMnCcVGs<\/a>. <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":17533,"menu_order":21,"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 146820, 146818\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine Volume of a Cone\",\"author\":\"James Sousa 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