{"id":68738,"date":"2021-01-22T14:59:27","date_gmt":"2021-01-22T20:59:27","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=68738"},"modified":"2026-03-25T10:55:07","modified_gmt":"2026-03-25T15:55:07","slug":"volume-and-surface-area-of-a-prism","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/volume-and-surface-area-of-a-prism\/","title":{"rendered":"Volume and Surface Area of a Prism"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_0bMI1faXgeA\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_0bMI1faXgeA\" data-source-videoID=\"0bMI1faXgeA\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Volume and Surface Area of a Prism Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Volume and Surface Area of a Prism\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_0bMI1faXgeA:hover {cursor:pointer;} img#videoThumbnailImage_0bMI1faXgeA {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1735-volume-and-surface-area-of-a-prism-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_0bMI1faXgeA\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_0bMI1faXgeA\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Volume and Surface Area of a Prism\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_0bMI1faXgeA\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_0bMI1faXgeA\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_0bMI1faXgeA\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction SQp_Function() {\n  var x = document.getElementById(\"SQp\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"SQp_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"SQp\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Key_Terms_to_Remember\" class=\"smooth-scroll\">Key Terms to Remember<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Rectangular_Prism_Volume_Formula\" class=\"smooth-scroll\">Rectangular Prism Volume Formula<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Rectangular_Prism_Surface_Area_Formula\" class=\"smooth-scroll\">Rectangular Prism Surface Area Formula<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_2\" class=\"smooth-scroll\">Example #2<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_3\" class=\"smooth-scroll\">Example #3<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Volume_and_Surface_Area_of_a_Prism_Practice_Questions\" class=\"smooth-scroll\">Volume and Surface Area of a Prism Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label><a href=\"https:\/\/www.mometrix.com\/academy\/cube-rectangular-prism-calculator\/\" target=\"none\" style=\"margin: 0 auto;\"><span class=\"accordion_calculator_button\">Calculator<\/span><\/a><\/p>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video on finding the volume and surface area of a prism!<\/p>\n<h2><span id=\"Key_Terms_to_Remember\" class=\"m-toc-anchor\"><\/span>Key Terms to Remember<\/h2>\n<p>\nBefore we jump into how to find the volume and surface area of a prism, let\u2019s go over a few key terms that we will see in our formulas. The first word we need to define is base. The bases of a prism are the two unique sides that the prism is named for. For example, if you have a <strong>hexagonal prism<\/strong>, the bases are the two hexagons on either end of the prism. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-68744\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/prism.png\" alt=\"Prism\" width=\"200\" height=\"200\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/prism.png 200w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/prism-150x150.png 150w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/><\/p>\n<p>The other word that will come up regularly in our formulas is height. Height is important to distinguish because it is different than the height used in some of our area formulas. The height of a prism is the length of an edge between the two bases. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-68741\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/prism1-e1611344273465.png\" alt=\"Prism\" width=\"246\" height=\"264\" \/><br \/>\nAnd finally, I want to review the word <strong>regular<\/strong>. Remember, regular in terms of polygons means that each side of the polygon has the same length.<\/p>\n<p>Now that we have gone over some of our key terms, let\u2019s look at our two formulas.<\/p>\n<h2><span id=\"Rectangular_Prism_Volume_Formula\" class=\"m-toc-anchor\"><\/span>Rectangular Prism Volume Formula<\/h2>\n<p>\nTo find the volume of a prism, multiply the area of the prism\u2019s base times its height. This is written as \\(V=Bh\\). Notice that big B stands for area of the base. It is important that you capitalize this B because otherwise it simply means base. We see this in the formula for the area of a triangle, \\(\\frac{1}{2}bh\\).<\/p>\n<h2><span id=\"Rectangular_Prism_Surface_Area_Formula\" class=\"m-toc-anchor\"><\/span>Rectangular Prism Surface Area Formula<\/h2>\n<p>\nThe formula for the surface area of a prism is \\(SA=2B+ph\\), where \\(B\\), again, stands for the area of the base, \\(p\\) represents the perimeter of the base, and \\(h\\) stands for the height of the prism.<\/p>\n<p>Now that we know what the formulas are, let\u2019s look at a few example problems using them.<\/p>\n<h2><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h2>\n<p>\nFind the volume and surface area of this rectangular prism.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-one.webp\" alt=\"\" width=\"525\" height=\"258\" class=\"aligncenter size-full wp-image-201791\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-one.webp 2100w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-one-300x147.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-one-1024x503.webp 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-one-768x377.webp 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-one-1536x755.webp 1536w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-one-2048x1006.webp 2048w\" sizes=\"auto, (max-width: 525px) 100vw, 525px\" \/><\/p>\n<p>Let\u2019s start with volume.<\/p>\n<p>\\(V=Bh\\)<br \/>\n&nbsp;<br \/>\nSince \\(B\\) stands for area of the base, we are going to substitute in the formula for area of a rectangle, length times width.<\/p>\n<p>So \\(V=lwh\\)<br \/>\n&nbsp;<br \/>\nNow, we are going to plug in our values.<\/p>\n<p>\\(V=4\\times 7\\times 13\\)<br \/>\n&nbsp;<br \/>\nWhen we multiply these out, this gives us \\(364 m^3\\). Remember, since we are multiplying by three dimensions, our units are cubed.<\/p>\n<p>Now let\u2019s look at our surface area.<\/p>\n<p>\\(SA=2B+ph\\)<br \/>\n&nbsp;<br \/>\nAgain, we are going to substitute in our formula for area of a rectangle, and we are also going to substitute in our formula for perimeter of a rectangle.<\/p>\n<p>\\(SA=2lw+2(l+w)h\\)<br \/>\n&nbsp;<br \/>\nNow we can plug in our known values.<\/p>\n<p>\\(SA=2(13\\times 7)+2(13+7)(4)\\)<br \/>\n&nbsp;<br \/>\nAnd we simplify to get our answer.<\/p>\n<p>\\(SA=2(91)+2(20)\u00d74\\)<br \/>\n\\(=182+160\\)<br \/>\n\\(=342\\text{ m}^2\\)<br \/>\n&nbsp;<br \/>\nThe surface area is 342 meters squared.<\/p>\n<p>Remember, with surface area, we are adding the areas of each face together, so we are only multiplying by two dimensions, which is why we square our units.<\/p>\n<h2><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h2>\n<p>\nLet\u2019s try another example.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-two.webp\" alt=\"\" width=\"386.24\" height=\"325.44\" class=\"aligncenter size-full wp-image-201794\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-two.webp 1207w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-two-300x253.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-two-1024x863.webp 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-two-768x647.webp 768w\" sizes=\"(max-width: 1207px) 100vw, 1207px\" \/><\/p>\n<p>Find the volume and surface area of this regular pentagonal prism.<\/p>\n<p>Let\u2019s start with our volume.<\/p>\n<p>\\(V=Bh\\)<br \/>\n&nbsp;<br \/>\nWe want to substitute in our formula for the area of a regular pentagon. This formula isn\u2019t common, so it\u2019s okay if you need to look it up. <\/p>\n<p>The area of a regular pentagon is found by \\(V=(\\frac{1}{2}pa)h\\)<\/p>\n<p>Now we can plug in our values. Remember, regular means that all the sides of the pentagon are congruent, so we can find our perimeter by multiplying our side value by 5.<\/p>\n<p>\\(V=(\\frac{1}{2}\\times 5\\times 5\\times 3)(14)\\)<br \/>\n&nbsp;<br \/>\nWhich, when we multiply this out, gives us 525 cubic feet.<\/p>\n<p>Now let\u2019s move on to surface area.<\/p>\n<p>\\(SA=2B+ph\\)<br \/>\n&nbsp;<br \/>\nFirst, we want to substitute in our formulas.<\/p>\n<p>\\(SA=2(\\frac{1}{2}pa)+(5s)h\\)<br \/>\n&nbsp;<br \/>\nAnd now we plug in our values.<\/p>\n<p>\\(SA=2(\\frac{1}{2}\\times 5\\times 5\\times 3)+(5\\times 5)(14)\\)<br \/>\n\\(=75+350\\)<br \/>\n\\(=425\\text{ ft}^2\\)<br \/>\n&nbsp;<br \/>\nThe surface area of our prism is 425 square feet.<\/p>\n<h2><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h2>\n<p>\nLet\u2019s look at one more example, but this time I want you to try it on your own.<\/p>\n<p>Find the volume and surface area of this regular triangular prism.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-three.webp\" alt=\"\" width=\"429.4\" height=\"339.72\" class=\"aligncenter size-full wp-image-201800\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-three.webp 1130w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-three-300x237.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-three-1024x810.webp 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/volume-and-surface-area-of-a-prism-example-three-768x608.webp 768w\" sizes=\"(max-width: 1130px) 100vw, 1130px\" \/><\/p>\n<p>Pause the video and see if you can come up with the answers on your own. Then check them with mine.<\/p>\n<p>Ready to check?<\/p>\n<p>Let\u2019s look at volume first.<\/p>\n<p>\\(V=Bh\\)<br \/>\n&nbsp;<br \/>\nWe substitute in our formula for area of a triangle.<\/p>\n<p>\\(V=(\\frac{1}{2}bh_T)h\\)<br \/>\n&nbsp;<br \/>\nNotice that I put a \\(T\\) on the height of the triangle to distinguish it from the height of the prism.<\/p>\n<p>Now let\u2019s plug in our values.<\/p>\n<p>\\(V=(\\frac{1}{2}\\times 12\\times 5)(25)=750 in^3\\)<br \/>\n&nbsp;<br \/>\nThe volume of our triangular prism is 750 cubic inches.<\/p>\n<p>Now onto surface area.<\/p>\n<p>\\(SA=2B+ph\\)<br \/>\n&nbsp;<br \/>\nFirst, substitute in your formulas.<\/p>\n<p>\\(SA=2(\\frac{1}{2}bh)+(3s)h\\)<br \/>\n&nbsp;<br \/>\nWe can use 3s for the perimeter because it is a regular triangle, or an equilateral triangle, so all the sides are the same length.<\/p>\n<p>Now, we plug in our values and solve.<\/p>\n<p>\\(SA=2(\\frac{1}{2}\\times 5\\times 12)+(3\\times 12)(25)=60+900=960\\text{ in}^2\\)<br \/>\n&nbsp;<br \/>\nThe surface area of our triangular prism is 960 square inches.<\/p>\n<p>And that\u2019s all there is to it! I hope this review of the volume and surface area of prisms was helpful. Thanks for watching and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Volume_and_Surface_Area_of_a_Prism_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Volume and Surface Area of a Prism Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFind the volume of the rectangular prism shown below:<\/p>\n<p><img decoding=\"async\" class=\"wp-image-111120 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-1.png\" alt=\"12 ft by 8 ft by 3 ft rectangular prism\" width=\"458.25\" height=\"237.75\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-1.png 1368w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-1-300x156.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-1-1024x531.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-1-768x399.png 768w\" sizes=\"(max-width: 1368px) 100vw, 1368px\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">288 ft<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-1-2\">300 ft<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-1-3\">60 ft<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-1-4\">23 ft<sup>3<\/sup><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To find the volume of a rectangular prism, use the formula \\(V = Bh\\), where \\(B\\) is the area of the base and \\(h\\) is the height of the prism.<\/p>\n<p>Because the base is a rectangle, the area of the base is found using \\(B = lw\\), where \\(l\\) is the length and \\(w\\) is the width.<\/p>\n<p>In this problem, the base has dimensions 8 feet by 3 feet, and the height of the prism is 12 feet. Substitute these values into the formula:<\/p>\n<p style=\"text-align: center\">\\(V = (8 \\times 3)(12)\\)<\/p>\n<p>First, find the area of the base:<\/p>\n<p style=\"text-align: center\">\\(V = (24)(12)\\)<\/p>\n<p>Now multiply to find the volume:<\/p>\n<p style=\"text-align: center\">\\(V = 288 \\text{ ft}^3\\)<\/p>\n<p>So, the volume of the rectangular prism is 288 cubic feet.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFind the surface area of the rectangular prism shown below:<\/p>\n<p><img decoding=\"async\" class=\"wp-image-111117 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-2.png\" alt=\"15 cm by 7 cm by 5 cm rectangular prism\" width=\"459.75\" height=\"238.5\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-2.png 1368w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-2-300x156.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-2-1024x531.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-2-768x399.png 768w\" sizes=\"(max-width: 1368px) 100vw, 1368px\" \/> <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">525 cm<sup>2<\/sup><\/div><div class=\"PQ\"  id=\"PQ-2-2\">525 cm<sup>2<\/sup><\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">430 cm<sup>2<\/sup><\/div><div class=\"PQ\"  id=\"PQ-2-4\">430 cm<sup>2<\/sup><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To find the surface area of a rectangular prism, use the formula \\(SA = 2B + ph\\), where \\(B\\) is the area of the base, \\(p\\) is the perimeter of the base, and \\(h\\) is the height of the prism.<\/p>\n<p>Because the base is a rectangle, the area of the base is \\(B = lw\\), and the perimeter of the base is \\(p = 2(l + w)\\).<\/p>\n<p>For this problem, the base has dimensions 7 cm by 5 cm, and the height of the prism is 15 cm. Substitute these values into the formula:<\/p>\n<p style=\"text-align: center\">\\(SA = 2(7 \\times 5) + 2(7 + 5)(15)\\)<\/p>\n<p>First, simplify inside the parentheses:<\/p>\n<p style=\"text-align: center\">\\(SA = 2(35) + 2(12)(15)\\)<\/p>\n<p>Now multiply:<\/p>\n<p style=\"text-align: center\">\\(SA = 70 + 360\\)<\/p>\n<p>Finally, add:<\/p>\n<p style=\"text-align: center\">\\(SA = 430 \\text{ cm}^2\\)<\/p>\n<p>So, the surface area of the rectangular prism is 430 square centimeters.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA pentagonal prism has a height of 10 meters, a perimeter of 30 meters, and its apothem is 2 meters. Based on this information, find the volume of the pentagonal prism. <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">600 m<sup>3<\/sup><\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">300 m<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-3-3\">42 m<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-3-4\">30 m<sup>3<\/sup><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To find the volume of a prism, use the formula \\(V = Bh\\), where \\(V\\) stands for volume, \\(B\\) stands for the area of the base of the prism, and \\(h\\) stands for its height.<\/p>\n<p>Since this is a pentagonal prism, substitute the area formula of a pentagon for \\(B\\). The area of the base of the prism is equal to \\(B = \\tfrac{1}{2}pa\\), which is one-half of the perimeter of the pentagon times the apothem.<\/p>\n<p>From here, replace the variables with their corresponding values given in the problem:<\/p>\n<p style=\"text-align: center\">\\(V = \\left(\\tfrac{1}{2} \\times 30 \\times 2\\right)(10)\\)<\/p>\n<p>Next, simplify expressions in parentheses. Since \\(\\tfrac{1}{2} \\times 30 \\times 2 = 30\\), rewrite the equation using this product:<\/p>\n<p style=\"text-align: center\">\\(V = (30)(10)\\)<\/p>\n<p>Multiply 30 by 10 to find the volume of the prism:<\/p>\n<p style=\"text-align: center\">\\(V = 300 \\text{ m}^3\\)<\/p>\n<p>So, the volume of the pentagonal prism is 300 cubic meters.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nJake works for an outdoor recreation company that makes canvas tents for camping. Jake\u2019s job is to purchase the exact amount of canvas needed for each tent. The tent he is working on is in the shape of a congruent triangular prism, as shown below. It has a base of 13 feet, a height of 8 feet, and a depth of 25 feet. Based on this information, find the surface area of the tent to figure out how much canvas Jake needs to purchase. <\/p>\n<p><img decoding=\"async\" class=\"wp-image-111111 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-3a.png\" alt=\"canvas tent resembling triangular prism\" width=\"455.6\" height=\"333.2\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-3a.png 1266w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-3a-300x219.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-3a-1024x749.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Volume-and-Surface-Area-of-Prisms-SQ-3a-768x562.png 768w\" sizes=\"(max-width: 1266px) 100vw, 1266px\" \/> <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">738 ft<sup>2<\/sup><\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\">944.5 ft<sup>2<\/sup><\/div><div class=\"PQ\"  id=\"PQ-4-3\">2,600 ft<sup>2<\/sup><\/div><div class=\"PQ\"  id=\"PQ-4-4\">1,079 ft<sup>2<\/sup><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To find the total surface area, we need to add the areas of all five faces.<\/p>\n<p>The area of one triangle is \\(\\frac{1}{2} bh\\):<\/p>\n<p style=\"text-align: center\">\\(A =\\frac{1}{2}\\times 13 \\times 8 = 52 \\mathrm{\\:ft}^2\\)<\/p>\n<p>Since there are two triangles, we turn 52 ft<sup>2<\/sup> into 104 ft<sup>2<\/sup>.<\/p>\n<p>The floor of the tent is a rectangle with the base of the triangle (13 ft) and the depth of the tent (25 ft), so the area of the floor is 325 ft<sup>2<\/sup>:<\/p>\n<p style=\"text-align: center\">\\(A = 13 \\times 25 = 325 \\mathrm{\\:ft}^2\\)<\/p>\n<p>Assuming the tent is an isosceles triangular prism, the height (8) bisects the base (13) into two segments of 6.5. We find the slant height (\\(s\\)) using:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(s^2 = 8^2 + 6.5^2\\)<br \/>\n\\(s^2 = 64 +42.25=106.25\\)<br \/>\n\\(s \\approx 10.31\\mathrm{\\:ft}\\)\n<\/p>\n<p>Now, calculate the area of the two slanted sides:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(\\text{Area} = 2 \\times (10.31 \\times 25) \\approx 515.5 \\mathrm{\\:ft}^2\\)<\/p>\n<p>Therefore, the total surface area of the tent is:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(104 + 325 + 515.5 =  944.5 \\mathrm{\\:ft}^2\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA wooden wedge in the shape of a triangular prism is being used as a doorstop. Its base is 2 inches, its height is 1.5 inches, and its depth is 4 inches. Based on this information, what is the volume of the doorstop? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">12 in<sup>2<\/sup><\/div><div class=\"PQ\"  id=\"PQ-5-2\">16 in<sup>3<\/sup><\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">6 in<sup>3<\/sup><\/div><div class=\"PQ\"  id=\"PQ-5-4\">8 in<sup>3<\/sup><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To find the volume of a prism, use the formula \\(V = Bh\\), where \\(V\\) stands for volume, \\(B\\) stands for the area of the base of the prism, and \\(h\\) stands for its height.<\/p>\n<p>Since this is a triangular prism, substitute the area formula of a triangle for \\(B\\). The base of the prism is equal to \\(B = \\tfrac{1}{2}bh\\), which is one-half of the base of the triangle times its height.<\/p>\n<p>From here, replace the variables with the corresponding values given in the problem:<\/p>\n<p style=\"text-align: center\">\\(V = \\left(\\tfrac{1}{2} \\times 2 \\times 1.5\\right)(4)\\)<\/p>\n<p>Next, simplify expressions in parentheses. Since \\(\\tfrac{1}{2} \\times 2 \\times 1.5 = 1.5\\), rewrite the equation using this product:<\/p>\n<p style=\"text-align: center\">\\(V = (1.5)(4)\\)<\/p>\n<p>Multiply 1.5 by 4 to find the volume of the prism:<\/p>\n<p style=\"text-align: center\">\\(V = 6 \\text{ in}^3\\)<\/p>\n<p>So, the volume of the triangular prism is 6 cubic inches.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/geometry\/\">Return to Geometry Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Geometry Videos<\/p>\n","protected":false},"author":1,"featured_media":100738,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-68738","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-volume-and-surface-area","7":"page_type-video","8":"content_type-practice-questions","9":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/68738","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=68738"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/68738\/revisions"}],"predecessor-version":[{"id":279007,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/68738\/revisions\/279007"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100738"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=68738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}