{"id":276405,"date":"2023-02-13T10:30:36","date_gmt":"2023-02-13T10:30:36","guid":{"rendered":"https:\/\/www.storyofmathematics.com\/?page_id=276405"},"modified":"2023-04-10T08:39:46","modified_gmt":"2023-04-10T08:39:46","slug":"surface","status":"publish","type":"page","link":"https:\/\/www.storyofmathematics.com\/glossary\/surface\/","title":{"rendered":"Surface | Definition &#038; Meaning"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"276405\" class=\"elementor elementor-276405\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-386a5e0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"386a5e0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-10661b0\" data-id=\"10661b0\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-61f1d92 elementor-widget elementor-widget-text-editor\" data-id=\"61f1d92\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<h1>Surface|Definition &amp; Meaning<\/h1>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-609e165 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"609e165\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fcaae43\" data-id=\"fcaae43\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-170da3d elementor-widget elementor-widget-text-editor\" data-id=\"170da3d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<h2><b>Definition<\/b><\/h2><p><span style=\"font-weight: 400\">The outer boundary of any three-dimensional object is called the <\/span><b>surface<\/b><span style=\"font-weight: 400\">. It may be flat as in a pyramid or curved as in a sphere or cylinder. By definition, it is 2D and therefore has no thickness but does have area. The area occupied by the surface is generally called the surface area.<\/span><\/p><!--[if lt IE 9]><script>document.createElement('audio');<\/script><![endif]-->\n<audio class=\"wp-audio-shortcode\" id=\"audio-276405-1\" preload=\"none\" style=\"width: 100%;\" controls=\"controls\"><source type=\"audio\/mpeg\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/m24-surface.mp3?_=1\" \/><a href=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/m24-surface.mp3\">https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/m24-surface.mp3<\/a><\/audio>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b6534cd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b6534cd\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6c32a12\" data-id=\"6c32a12\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b734f2d elementor-widget elementor-widget-text-editor\" data-id=\"b734f2d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div style=\"overflow-x: scroll;\">\n<p><span style=\"font-weight: 400;\">An object&#8217;s outermost layer.<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">It&#8217;s a 2-dimensional boundary that may be flat or curved&nbsp;<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">It has an area but still no thickness.&nbsp;<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Figure 1 below shows the surface of a cube.<\/span><\/p>\n<div id=\"attachment_276407\" style=\"width: 650px\" class=\"wp-caption aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" aria-describedby=\"caption-attachment-276407\" class=\"size-large wp-image-276407\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube-1024x512.png\" alt=\"\" width=\"640\" height=\"320\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube-1024x512.png 1024w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube-300x150.png 300w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube-768x384.png 768w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube-1536x768.png 1536w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube-600x300.png 600w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube-1298x649.png 1298w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-of-a-cube.png 1800w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><p id=\"caption-attachment-276407\" class=\"wp-caption-text\">Figure 1 &#8211; Surface of a cube.<\/p><\/div>\n<p><span style=\"font-weight: 400;\">A 2-dimensional point collection (flat surface), a 3-dimensional point collection with a curve cross-section (curved surface), or the border of any 3-dimensional solid. A surface is a continuous barrier that divides a 3-dimensional space into two sections. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">A sphere&#8217;s surface, for example, divides the interior from the exterior; a parallel to the horizontal divide the half-plane over it to the half-plane below it. Surfaces are frequently referred to by the name of the area they surround. However, a surface is fundamentally two-dimensional with an area, but the area it encloses has 3-dimensional with a volume.<\/span><\/p>\n<h2><b>Types of Surface<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">There are two types of surfaces<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Flat surface&nbsp;<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Curved surface<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Figure 2 below shows the types of surfaces.<\/span><\/p>\n<div id=\"attachment_276409\" style=\"width: 650px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" aria-describedby=\"caption-attachment-276409\" class=\"size-large wp-image-276409\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface-1024x512.png\" alt=\"\" width=\"640\" height=\"320\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface-1024x512.png 1024w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface-300x150.png 300w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface-768x384.png 768w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface-1536x768.png 1536w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface-600x300.png 600w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface-1298x649.png 1298w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Representation-of-different-types-of-surface.png 1800w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><p id=\"caption-attachment-276409\" class=\"wp-caption-text\">Figure 2 &#8211; Types of surfaces in a cylinder.<\/p><\/div>\n<h2><b>Flat Surface<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">The <\/span><b>flat surface <\/b><span style=\"font-weight: 400;\">is another name for a flat surface. Plane geometry is concerned with flat forms that may be drawn on paper, including squares, circles, and triangles. A flat figure or plane has two dimensions: width and length.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">A solid, as well as a 3D form, occupies space. The surface of a solid refers to the solid&#8217;s outer layer. So when the surface of the solid is a flat surface with really no depths or flatness, it is referred to as a flat surface. Every day, we notice a lot of flat items in our surroundings. For example, a book, desk, or dresser can have a flat surface.<\/span><\/p>\n<h2><b>Recognizing Flat Surfaces within 3D Shapes<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">If we look at various 3D blocks, cubes, pyramids, and prisms, we&#8217;ll see that they only have flat or plane surfaces. 3D figures and forms have flat, curved surfaces, such as cylindrical or cones. Furthermore, particular 3D objects, such as spheres, have no flat surfaces and just one curved surface. You can move an object with a flat surface. If an object has a curved surface, you can roll it along it.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">&nbsp;The cubes have six sides, each of which is a square. The cylinders have a curved surface plus two flat surfaces that are equal circles. A spherical one has only one curved surface.<\/span><\/p>\n<h2><b>Curved Surface<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">A <\/span><b>curved surface <\/b><span style=\"font-weight: 400;\">is indeed a rounded, non-flat surface. A curved surface can be found everywhere around an item. Such items have a single surface all the way around. Spheres are examples of things with curved surfaces all around.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Real-world examples of items with curved surfaces include balls, globes, eggs, pipes, domes, and so on.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">There are 3D forms that solely have flat surfaces. For example, a cube, cuboid, pyramid, and prism are all 3D geometries formed out of flat surfaces. They have squares, rectangles, triangles, and parallelograms as their surfaces. They don&#8217;t have any curved surfaces. Boxes, cubes, pyramids, prisms, bricks, and other 3D structures with curved surfaces are not examples.<\/span><\/p>\n<h2><b>Area of a Surface<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">A three-dimensional object&#8217;s surface area is the sum of its faces. Real-world applications of surface areas include wrapping, painting, and constructing objects to get the most excellent potential design. An object&#8217;s surface area is the total area that all of its surfaces occupy.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">There are two groups for the surface area:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Total surface area<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">The surface area that is laterally oriented or curved<\/span><\/li>\n<\/ul>\n<h2><b>Surface Area Types<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">Three-dimensional forms can have either a total surface area or a curved\/lateral surface area. Although the curved and lateral area only comprises the area of either the lateral faces of the forms, the surface area encompasses the space of all of the faces of a shape. To understand the distinction between the total and curved surface area, look at the cylinder shown below.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Figure 3 below shows the area of the cylinder.<\/span><\/p>\n<div id=\"attachment_276408\" style=\"width: 650px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" aria-describedby=\"caption-attachment-276408\" class=\"size-large wp-image-276408\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder-1024x512.png\" alt=\"\" width=\"640\" height=\"320\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder-1024x512.png 1024w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder-300x150.png 300w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder-768x384.png 768w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder-1536x768.png 1536w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder-600x300.png 600w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder-1298x649.png 1298w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/01\/Surface-area-of-a-cylinder.png 1800w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><p id=\"caption-attachment-276408\" class=\"wp-caption-text\">Figure 3 &#8211; Surface area of a cylinder.<\/p><\/div>\n<h2><b>Prism Surface Area<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">A <strong>prism<\/strong> is a 3D solid object made up of two congruent foundations, a polygon, and two congruent lateral sides, rectangular in shape.<\/span><\/p>\n<p><span style=\"font-weight: 400;\"> Only a prism has two distinct areas: its lateral total surface area and the total lateral surface. Whereas the lateral surface area total of just a prism is the combination of both the regions of each of its lateral faces, the external overall surface area of a prism is the product of something like the main area and the area among its bases.<\/span><\/p>\n<p><b>The prism&#8217;s lateral surface area<\/b><span style=\"font-weight: 400;\"> = height \u00d7 base perimeter&nbsp;&nbsp;<\/span><\/p>\n<p><b>The prism&#8217;s total surface area<\/b><span style=\"font-weight: 400;\"> = Lateral surface area of prism + area of the two bases = <\/span><span style=\"font-weight: 400;\">Lateral surface area + (2 \u00d7 Base Area) = (Base perimeter \u00d7 height) +&nbsp; (2 \u00d7 Base Area).<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Based on how the base of prisms is shaped, there are seven different varieties of prisms. The formulae used to calculate the overall surface area of a prism varies, just as the bases of various prisms do.<\/span><\/p>\n<h2><b>Examples of Surface<\/b><\/h2>\n<p><span style=\"font-weight: 400;\">Some of the examples of surfaces are listed below<\/span><\/p>\n<h3><b>Example 1<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">What is the ice cream cone&#8217;s surface area if its radius and slant height are 5 and 8 inches, respectively?<\/span><\/p>\n<h3><b>Solution<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">Given: slant height equals 8 inches, radius = 5 inches.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Cone&#8217;s surface area is equal to pi(r + l).<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">= $\\pi$ \u00d7 5(5 + 8)<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">= 3.14 \u00d7 5 \u00d7 13<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">= 204.1 square inches<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The cone has a surface area of 204.1 inches square.<\/span><\/p>\n<h3><b>Example 2<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">How big is its surface area if a cube has 5 inches on each side?<\/span><\/p>\n<h3><b>Solution<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">Given that the cube&#8217;s sides are 5 inches long.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">A cube&#8217;s surface area equals 6a<sup>2<\/sup>.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">a =5 inches (given)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">When we replace the quantities in the equation, we obtain,<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">= 6(5)<sup>2<\/sup><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">= 6 (25)&nbsp;<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">=&nbsp; 150 square inches<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The cube&#8217;s 54 inches2 of surface area.<\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">All Images are made using GeoGebra.<\/span><\/i><\/p>\n<h4 style=\"text-align: center;\"><a href=\"https:\/\/www.storyofmathematics.com\/glossary\/surd\/\" target=\"_blank\" rel=\"noopener\">Surd Definition<\/a> &lt; <a href=\"https:\/\/www.storyofmathematics.com\/glossary\/\" target=\"_blank\" rel=\"noopener\">Glossary Index<\/a> &gt; <a href=\"https:\/\/www.storyofmathematics.com\/glossary\/survey\/\" target=\"_blank\" rel=\"noopener\">Survey Definition<\/a><\/h4>\n<\/div>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Surface|Definition &amp; Meaning Definition The outer boundary of any three-dimensional object is called the surface. It may be flat as in a pyramid or curved as in a sphere or cylinder. By definition, it is 2D and therefore has no thickness but does have area. The area occupied by the surface is generally called the [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":24,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"rank_math_lock_modified_date":false,"footnotes":""},"categories":[],"tags":[],"class_list":["post-276405","page","type-page","status-publish","hentry","entry"],"acf":[],"rankMath":{"parentDomain":"storyofmathematics.com","noFollowDomains":[],"noFollowExcludeDomains":[],"noFollowExternalLinks":false,"featuredImageNotice":"The featured image should be at least 200 by 200 pixels to be picked up by Facebook and other social media sites.","pluginReviewed":false,"postSettings":{"linkSuggestions":true,"useFocusKeyword":false},"frontEndScore":false,"postName":"surface","permalinkFormat":"https:\/\/www.storyofmathematics.com\/%pagename%\/","showLockModifiedDate":true,"assessor":{"focusKeywordLink":"https:\/\/www.storyofmathematics.com\/wp-admin\/edit.php?focus_keyword=%focus_keyword%&post_type=%post_type%","hasTOCPlugin":true,"primaryTaxonomy":false,"serpData":{"title":"","description":"The outer boundary of any three-dimensional object is called the surface. 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