{"id":2865,"date":"2023-04-26T10:00:07","date_gmt":"2023-04-26T04:30:07","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=2865"},"modified":"2023-04-26T09:26:40","modified_gmt":"2023-04-26T03:56:40","slug":"surface-area-volume-cube","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/surface-area-volume-cube\/","title":{"rendered":"How do you find the Surface Area and Volume of a Cube"},"content":{"rendered":"<h2><strong>How do you find the Surface Area and Volume of a Cube<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12824\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Cube.png\" alt=\"\" width=\"729\" height=\"433\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Cube.png 729w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Cube-300x178.png 300w\" sizes=\"(max-width: 729px) 100vw, 729px\" \/><\/p>\n<p>If the length of each edge of a cube is \u2018a\u2019 units, then<\/p>\n<ol>\n<li>Total surface area of the cube = 6a<sup>2<\/sup>\u00a0 sq. units.<\/li>\n<li>Lateral surface area = 4a<sup>2<\/sup><\/li>\n<li>Volume of the cube = a<sup>3<\/sup> cubic units<\/li>\n<li>Diagonal of the cube = a units.<\/li>\n<li>Length of its diagonals = a\u221a3<\/li>\n<li>Total length of its edges = 12a<\/li>\n<\/ol>\n<p><strong>Read More:<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/www.aplustopper.com\/surface-area-volume-cuboid\/\"><span style=\"font-weight: 400;\">Surface Area and Volume of a Cuboid<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/surface-area-sphere-hemisphere\/\"><span style=\"font-weight: 400;\">Surface Area of a Sphere and a Hemisphere<\/span><\/a><\/li>\n<\/ul>\n<h2><strong>Surface Area and Volume of a Cube Example Problems with Solutions<\/strong><\/h2>\n<p><strong>Example 1:<\/strong>\u00a0 \u00a0 If each edge (side) of a cube is 8 cm ; find its surface area and lateral surface area.<br \/>\n<strong>Solution: \u00a0 \u00a0\u00a0<\/strong>Given each side of the cube (a) = 8 cm<br \/>\n\u2234\u00a0\u00a0\u00a0 Its surface area = 6a<sup>2<\/sup>\u00a0= 6 \u00d7 8<sup>2<\/sup>\u00a0sq. cm<br \/>\n= 6 \u00d7 64 cm<sup>2<\/sup>\u00a0\u00a0= 384 cm<sup>2<\/sup><br \/>\nLateral surface area = 4a<sup>2<\/sup>\u00a0= 4 \u00d7 8<sup>2<\/sup>\u00a0sq. cm<br \/>\n= 4 \u00d7 64 cm<sup>2<\/sup>\u00a0\u00a0= 256 cm<sup>2<\/sup><\/p>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fCAB4xJgJaA?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Example 2:<\/strong>\u00a0 \u00a0 A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.<br \/>\n(i)\u00a0\u00a0 Which box has the greater lateral surface area and by how much ?<br \/>\n(ii)\u00a0 Which box has the smaller total surface area and by how much ?<br \/>\n<strong>Solution: \u00a0 \u00a0 <\/strong>(i)\u00a0\u00a0 For the cubical box :<br \/>\nEach edge = 10 cm i.e., a = 10 cm<br \/>\n\u2234\u00a0\u00a0\u00a0 Lateral surface area of the cubical box<br \/>\n= 4a<sup>2<\/sup>\u00a0= 4 \u00d7 10<sup>2<\/sup>\u00a0cm<sup>2<\/sup>\u00a0= 400 cm<sup>2<\/sup><br \/>\nFor the cuboidal box :<br \/>\n\u2113 = 12.5 cm, b = 10 cm &amp; h = 8 cm<br \/>\n\u2234\u00a0\u00a0\u00a0 Lateral surface area of the cuboidal box<br \/>\n= 2(\u2113 + b) \u00d7 h<br \/>\n= 2(12.5 + 10) \u00d7 8 cm<sup>2<\/sup><br \/>\n= 2 \u00d7 22.5 \u00d7 8 cm2 = 360 cm<sup>2<\/sup><br \/>\nClearly, cubical box has greater lateral surface area by<br \/>\n400 cm<sup>2<\/sup>\u00a0\u2013 360 cm<sup>2<\/sup>\u00a0\u00a0= 40 cm<sup>2<\/sup><br \/>\n(ii)\u00a0 Total surface area of the cubical box:<br \/>\n= 6 a<sup>2<\/sup>\u00a0\u00a0= 6 \u00d7 10<sup>2<\/sup>\u00a0sq. cm = 600 cm<sup>2<\/sup><br \/>\nTotal surface area of the cuboidal box<br \/>\n= 2(\u2113 \u00d7 b + b \u00d7 h + h \u00d7 \u2113)<br \/>\n= 2(12.5 \u00d7 10 +10 \u00d7 8 + 8 \u00d7 12.5) cm<sup>2<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 = 2(125 + 80 + 100) cm<sup>2<\/sup>\u00a0= 610 cm<sup>2<\/sup><br \/>\nClearly, cubical box has smaller surface area by<br \/>\n610 cm<sup>2<\/sup>\u00a0\u2013 600 cm<sup>2<\/sup>\u00a0= 10 cm<sup>2<\/sup><\/p>\n<p><strong>Example 3:<\/strong>\u00a0 \u00a0 Find the volume of a solid cube of side 12 cm. If this cube is cut into 8 identical cubes, find :<br \/>\n(i)\u00a0\u00a0 Volume of each small cube.<br \/>\n(ii)\u00a0 Side of each small cube.<br \/>\n(iii) Surface area of each small cube.<br \/>\n<strong>Solution: \u00a0 \u00a0\u00a0<\/strong>Since, side (edge) of the given solid cube = 12 cm.<br \/>\n\u2234\u00a0\u00a0\u00a0 Volume of given solid cube = (edge)<sup>3<\/sup><br \/>\n= (12 cm)<sup>3<\/sup>\u00a0= 1728 cm<sup>3<\/sup>\u00a0\u00a0 Ans.<br \/>\n(i)\u00a0\u00a0 As the given cube is cut into 8 identical cubes.<br \/>\n\u21d2\u00a0\u00a0 Vol. of 8 small cubes obtained<br \/>\n= Vol. of given cube = 1728 cm<sup>3<\/sup><br \/>\n\u21d2 \u00a0 Volume of each small cube<br \/>\n= \\(\\frac{{1728\\,\\,c{m^3}}}{8}\\)\u00a0= 216 cm<sup>3<\/sup><br \/>\n(ii)\u00a0 If edge (side) of each small cube = x cm<br \/>\n(edge)<sup>3<\/sup>\u00a0= Volume<br \/>\n\u21d2\u00a0\u00a0 x<sup>3<\/sup> = 216 = 6 \u00d7 6 \u00d7 6 = 6<sup>3 <\/sup>\u21d2\u00a0\u00a0 x = 6 cm<br \/>\n\u2234\u00a0\u00a0\u00a0 Side of each small cube = 6 cm<br \/>\n(iii) Surface area of each small cube<br \/>\n= 6 \u00d7 (edge)<sup>2<\/sup><br \/>\n= 6 \u00d7 (6 cm)<sup>2<\/sup>\u00a0= 216 cm<sup>2<\/sup><\/p>\n<p><strong>Example 4:<\/strong>\u00a0 \u00a0 A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute ? <strong>\u00a0<\/strong><br \/>\n<strong>Solution: \u00a0 \u00a0\u00a0<\/strong>Volume of water that flows through a river, canal or pipe, etc., in unit time<br \/>\n= \u00a0\u00a0 Area of cross-section \u00d7 Speed of water through it.<br \/>\nx km\/hr = x \u00d7 \\(\\frac{5}{{18}}m\/s\\)<br \/>\nReason : 1 km\/hr =\u00a0\\(\\frac{{1000m}}{{60 \\times 60\\,\\,\\,\\sec }} = \\frac{5}{{18}}m\/s\\)<br \/>\nSince, area of cross-section of the river<br \/>\n= Its depth \u00d7 its width<br \/>\n= 3m \u00d7 40m = 120 m<sup>2<\/sup><br \/>\nAnd, speed of flow of water through the river<br \/>\n= 2 km\/hr = 2 \u00d7\u00a0\\(\\frac{5}{{18}}m\/s = \\frac{5}{9}m\/s\\)<br \/>\n\u2234 Vol. of water that flows through it in 1 sec.<br \/>\n=\u00a0 Area of cross-section \u00d7 speed of water through it.<br \/>\n= 120 \u00d7\u00a0\\(\\frac{5}{9}{m^3} = \\frac{{200}}{3}{m^3}\\)<br \/>\n\u21d2\u00a0\u00a0 Vol. of water that flows through it in<br \/>\n1 min. (60 sec.)<br \/>\n=\u00a0\\(\\frac{{200}}{3} \\times 60\\,\\,{m^3}\\) = 4000 m<sup>3<\/sup><br \/>\n\u21d2\u00a0\u00a0 Vol. of water that will fall into the sea in a minute. = 4000 m<sup>3<\/sup><\/p>\n<p><strong>Example 5:<\/strong>\u00a0 \u00a0 The volume of a cube is numerically equal to its surface area. Find the length of its one side.<br \/>\n<strong>Solution: \u00a0 \u00a0\u00a0<\/strong>Let length of each side is a unit.<br \/>\nGiven: Volume of the cube = Surface area of the cube.<br \/>\n\u21d2\u00a0\u00a0 a<sup>3<\/sup>\u00a0= 6a<sup>2<\/sup>\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 \u21d2\u00a0\u00a0 a = 6<br \/>\n\u2234\u00a0\u00a0\u00a0 The length of one side of the cube = 6 cm<\/p>\n<p><strong>Example 6:<\/strong>\u00a0 \u00a0 A solid cuboid has square base and height<br \/>\n12 cm. If its volume is 768 cm<sup>3<\/sup>, find :<br \/>\n(i)\u00a0\u00a0 side of its square base.<br \/>\n(ii)\u00a0 surface area.<br \/>\n<strong>Solution: \u00a0 \u00a0\u00a0<\/strong>(i)\u00a0\u00a0 Let side of the square base be x cm<br \/>\ni.e., \u2113 = b = x cm<br \/>\n\u2113 \u00d7 b \u00d7 h = volume<br \/>\n\u21d2\u00a0\u00a0 x \u00d7 x \u00d7 12 = 768<br \/>\n[Given, height = 12 cm]<br \/>\n\u21d2\u00a0\u00a0 x<sup>2<\/sup>\u00a0= \\(\\frac{{768}}{{12}} = 64\\)\u00a0\u21d2 \u00a0\u00a0x = \u221a64cm = 8 cm.<br \/>\n\u2234\u00a0\u00a0\u00a0 Side of the square base = 8 cm<br \/>\n(ii)\u00a0 Now, \u2113 = 8 cm, b = 8 cm and h = 12 cm<br \/>\n\u2234 \u00a0Surface area = 2(\u2113 \u00d7 b + b \u00d7 h + h \u00d7 \u2113)<br \/>\n= 2(8 \u00d7 8 + 8 \u00d7 12 + 12 \u00d7 8) cm<sup>2<\/sup> = 512 cm<sup>2<\/sup><\/p>\n","protected":false},"excerpt":{"rendered":"<p>How do you find the Surface Area and Volume of a Cube If the length of each edge of a cube is \u2018a\u2019 units, then Total surface area of the cube = 6a2\u00a0 sq. units. Lateral surface area = 4a2 Volume of the cube = a3 cubic units Diagonal of the cube = a units. [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[1237,1235,1239,1238,1234,1236],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>How do you find the Surface Area and Volume of a Cube - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/surface-area-volume-cube\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How do you find the Surface Area and Volume of a Cube\" \/>\n<meta property=\"og:description\" content=\"How do you find the Surface Area and Volume of a Cube If the length of each edge of a cube is \u2018a\u2019 units, then Total surface area of the cube = 6a2\u00a0 sq. units. 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Lateral surface area = 4a2 Volume of the cube = a3 cubic units Diagonal of the cube = a units. 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