{"id":35389,"date":"2023-10-30T11:39:05","date_gmt":"2023-10-30T11:39:05","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=35389"},"modified":"2023-10-30T14:29:03","modified_gmt":"2023-10-30T14:29:03","slug":"diagonal-of-a-cube-formula-definition-derivation-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula","title":{"rendered":"Diagonal of a Cube Formula &#8211; Definition, Derivation, Examples"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-763fedf6-2404-4250-9291-10672c48bf9d\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula#0-what-is-the-diagonal-of-a-cube-formula>What Is the Diagonal of a Cube Formula?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula#3-derivation-of-diagonal-of-a-cube-formula>Derivation of Diagonal of a Cube Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula#4-uses-of-diagonal-of-a-cube-formula>Uses of Diagonal of a Cube Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula#7-solved-examples-on-diagonal-of-a-cube-formula>Solved Examples on Diagonal of a Cube Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula#8-practice-problems-on-diagonal-of-a-cube-formula>Practice Problems on Diagonal of a Cube Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula#9-frequently-asked-questions-about-the-diagonal-of-a-cube-formula>Frequently Asked Questions about the Diagonal of a Cube Formula<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-the-diagonal-of-a-cube-formula\">What Is the Diagonal of a Cube Formula?<\/h2>\n\n\n\n<p><strong>The formula to calculate the length of diagonal of a cube (body diagonal) is <\/strong><strong>3<\/strong><strong> times the length of one of its sides.<\/strong> In mathematical terms, if &#8220;s&#8221; represents the length of a side, then the diagonal &#8220;d&#8221; can be expressed as $d = s\\sqrt{3}$.\u00a0<\/p>\n\n\n\n<p>A <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/cube\">cube<\/a> is a <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/3-dimensional\">three-dimensional<\/a> solid figure with 6 congruent square faces, 8 vertices, and 12 equal edges. All its edges are equal in length. This indicates that a cube has equal length, breadth, and height and that every one of its sides is a square.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"507\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/cube-dimensions.png\" alt=\"Cube dimensions\" class=\"wp-image-35394\" title=\"Cube dimensions\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/cube-dimensions.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/cube-dimensions-300x245.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>The diagonal of a cube is the line segment that connects any pair of non-adjacent sides of the cube. The main diagonal of a cube is the one that goes through the center of the cube; the face diagonal of a cube is not its primary diagonal.&nbsp;<\/p>\n\n\n\n<p>There are two kinds of diagonals of a cube.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Face Diagonal of a Cube<\/strong><\/li>\n<\/ul>\n\n\n\n<p>There are six square-shaped faces on a cube. The non-adjacent vertices of each face are connected by two diagonals. Therefore, there are actually 12 face diagonals.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"621\" height=\"441\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/face-diagonal-of-a-cube.png\" alt=\"Face diagonal of a cube\" class=\"wp-image-35395\" title=\"Face diagonal of a cube\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/face-diagonal-of-a-cube.png 621w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/face-diagonal-of-a-cube-300x213.png 300w\" sizes=\"auto, (max-width: 621px) 100vw, 621px\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Body Diagonal of a Cube<\/strong><\/li>\n<\/ul>\n\n\n\n<p>The body diagonals connect the two opposite vertices of the cube by passing through the internal structure of the cube. As a result, there are 4 diagonals crossing across the body. It is also known as \u201cspace diagonal,\u201d or \u201cinternal diagonal,\u201d or \u201cmain diagonal.\u201d<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"444\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube.png\" alt=\"Body diagonal of a cube\" class=\"wp-image-35396\" title=\"Body diagonal of a cube\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube-300x215.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/find-the-volume-using-unit-cubes\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geo_meas_vol_unit_cubes_2_pt.png\" alt=\"Find the Volume using Unit Cubes Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Find the Volume using Unit Cubes Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/find-volume-using-the-formula\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geo_meas_vol_formula_pt.png\" alt=\"Find Volume using the Formula Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Find Volume using the Formula Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-cones-and-cubes\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_cones_cubes_pt.png\" alt=\"Identify Cones and Cubes Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Cones and Cubes Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/make-a-number-using-cubes-up-to-5\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_make_num_w5_pt.png\" alt=\"Make a Number Using Cubes (Up to 5) Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Make a Number Using Cubes (Up to 5) Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/making-a-number-up-to-10-using-cubes\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_make_num_w10_pt.png\" alt=\"Making a Number (Up to 10) Using Cubes Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Making a Number (Up to 10) Using Cubes Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"1-diagonal-of-a-cube-formula\">Diagonal of a Cube Formula<\/h2>\n\n\n\n<p>Let\u2019s see the formulas to find the diagonal of a cube.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"wj-table-class\"><thead><tr><th><strong>Type of Diagonal<\/strong><\/th><th><strong>Diagram<\/strong><\/th><th><strong>Formula<\/strong><\/th><\/tr><\/thead><tbody><tr><td>Face Diagonal<\/td><td><img loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"194\" class=\"wp-image-35397\" style=\"width: 250px;\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/face-diagonal-of-a-cube-visual.png\" alt=\"Face diagonal of a cube - visual\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/face-diagonal-of-a-cube-visual.png 308w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/face-diagonal-of-a-cube-visual-300x233.png 300w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/td><td><br>$d = \\sqrt{2a}$<br>where \u201ca\u201d is the length of<br>the side of a cube.<\/td><\/tr><tr><td>Body Diagonal<\/td><td><img loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"200\" class=\"wp-image-35398\" style=\"width: 250px;\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube-visual.png\" alt=\"Body diagonal of a cube - visual\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube-visual.png 309w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube-visual-300x240.png 300w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/>\u00a0<\/td><td><br><br>$d = \\sqrt{3a}$<br>where \u201ca\u201d is the length of<br>the side of a cube.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Example 1:<\/strong> Calculate the length of the face diagonal of a cube whose each side is 8 inches long.<\/p>\n\n\n\n<p>Length of the side of the cube = a = 8 inches<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= \\sqrt{2a}$<\/p>\n\n\n\n<p>$\\Rightarrow d = \\sqrt{2} \\times 8$<\/p>\n\n\n\n<p>$\\Rightarrow d = 1.41 \\times 8$<\/p>\n\n\n\n<p>$\\Rightarrow d = 11.28$ inches<\/p>\n\n\n\n<p><strong>Example 2:<\/strong> Calculate the length of the body diagonal of a cube with side-length of 9 inches.<\/p>\n\n\n\n<p>Length of the side of the cube = a = 9 inches<\/p>\n\n\n\n<p>Length of body diagonal of the cube $= \\sqrt{3a}$<\/p>\n\n\n\n<p>$d = \\sqrt{3} \\times 9$<\/p>\n\n\n\n<p>$d = 1.73 \\times 9$<\/p>\n\n\n\n<p>Length of body diagonal of the cube $= 15.57$ inches<\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/cylinders-and-cubes\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/cylinders-and-cubes.jpeg\" alt=\"Cylinders and Cubes Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/worksheets\">More Worksheets<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"2-number-of-diagonals-in-a-cube\">Number of Diagonals in a Cube<\/h2>\n\n\n\n<p>We know that a cube has 6 faces. Each face has 2 diagonals connecting their non-adjacent sides.&nbsp;<\/p>\n\n\n\n<p>Number of face diagonals $= (2 \\times 6) = 12$\u00a0<\/p>\n\n\n\n<p>Additionally, there are 4 diagonals running across the body of the cube.&nbsp;<\/p>\n\n\n\n<p>Number of body diagonals $= 4$<\/p>\n\n\n\n<p>Therefore, the total no. of diagonals in a cube<strong> <\/strong>can be calculated as&nbsp;<\/p>\n\n\n\n<p>$12 + 4 = 16$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-derivation-of-diagonal-of-a-cube-formula\">Derivation of Diagonal of a Cube Formula<\/h2>\n\n\n\n<p>The formula for the diagonal of a cube is calculated using the Pythagoras theorem. The cube diagonal formula is usually referred to as the body diagonal formula.&nbsp;<\/p>\n\n\n\n<p>Consider a cube whose each side length is \u201ca\u201d units. If we draw any face diagonal, we can form a right-angled triangle with legs of \u201ca\u201d units and face diagonal as the hypotenuse.<\/p>\n\n\n\n<p>Face diagonal $= \\sqrt{a^{2} + a^{2}} = \\sqrt{2a^{2}} = \\sqrt{2a}$ units\u00a0<\/p>\n\n\n\n<p>Similarly, we will get a right-angled triangle with the body diagonal as the hypotenuse, side of the cube as one leg, and the face diagonal as the other leg.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"573\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube-formula.png\" alt=\"Body diagonal of a cube formula\" class=\"wp-image-35399\" title=\"Body diagonal of a cube formula\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube-formula.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/body-diagonal-of-a-cube-formula-300x277.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>Side length = a&nbsp;<\/p>\n\n\n\n<p>face diagonal $= \\sqrt{2a}$<\/p>\n\n\n\n<p>By Pythagoras theorem, we get<\/p>\n\n\n\n<p>$a^{2} + (\\sqrt{2a})^{2} = d^{2}$<\/p>\n\n\n\n<p>$\\Rightarrow a^{2} + 2a^{2} = d^{2}$<\/p>\n\n\n\n<p>$\\Rightarrow 3a^{2} = d^{2}$<\/p>\n\n\n\n<p>$\\Rightarrow d = \\sqrt{3a^{2}}$<\/p>\n\n\n\n<p>$\\Rightarrow d = \\sqrt{3a}$<\/p>\n\n\n\n<p>Length of the Body diagonal $= d = \\sqrt{3a}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-uses-of-diagonal-of-a-cube-formula\">Uses of Diagonal of a Cube Formula<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Determining Cube Size<\/strong>: The formula helps figure out how long the diagonal is within a cube, which is like measuring the longest diagonal in a rectangular box.<\/li>\n\n\n\n<li><strong>Visualizing 3D Shapes<\/strong>: Understanding cube diagonals gives you a way to imagine how lines can go from one corner of the cube to the farthest corner, helping you see how 3D shapes fit together.<\/li>\n\n\n\n<li><strong>Understanding Big Structures<\/strong>: Architects and builders use these diagonals to make sure big structures, like bridges or buildings, are strong and stable when they have cube-like parts.<\/li>\n\n\n\n<li><strong>Creating Cube Designs<\/strong>: If you want to build something using cubes, like a robot or a piece of furniture, knowing how diagonals work will help you make sure everything fits together just right.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-facts-about-the-diagonal-of-a-cube-formula\">Facts about the Diagonal of a Cube Formula<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li>The diagonal is the longest line that can be drawn within a cube, connecting two opposite corners (vertices).<\/li>\n\n\n\n<li>If you need to travel from one corner of a cube to its opposite corner, the body diagonal (space diagonal)&nbsp; is the shortest path.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we explored the formulas for calculating the face and body diagonals of a cube, crucial for determining distances between vertices. These concepts are pivotal in geometry and spatial reasoning. For enhanced understanding, let&#8217;s work through examples and engage in MCQs.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-solved-examples-on-diagonal-of-a-cube-formula\">Solved Examples on Diagonal of a Cube Formula<\/h2>\n\n\n\n<p><strong>Example 1: Find the face diagonal of a cube with sides of 7 units.&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Length of the side of the cube = 7 units<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= \\sqrt{2a}$<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= \\sqrt{2} \\times 7$<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= 1.41 \\times 7$<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= 9.87$ units<\/p>\n\n\n\n<p><strong>Example 2: What will be the body diagonal of a cube with sides of 11 inches?<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Length of the side of the cube = 11 inches<\/p>\n\n\n\n<p>Length of body diagonal of the cube $= \\sqrt{3a}$<\/p>\n\n\n\n<p>Length of body diagonal of the cube $= \\sqrt{3} \\times 11$<\/p>\n\n\n\n<p>Length of body diagonal of the cube $= 1.73 \\times 11$<\/p>\n\n\n\n<p>Length of body diagonal of the cube =19.03 inches<\/p>\n\n\n\n<p><strong>Example 3: The body diagonal of a cube is 4\u221a3 units. What will be the length side of the cube and the length of the face diagonal of the cube?<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Length of body diagonal of the cube $= \\sqrt{3a}$<\/p>\n\n\n\n<p>$\\sqrt{3a} = 4\\sqrt{3}$<\/p>\n\n\n\n<p>Length of side of the cube =4 units<\/p>\n\n\n\n<p>Length of the side of the cube =4 units<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= \\sqrt{2a}$<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= \\sqrt{2} \\times 4$<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= 1.41 \\times 4$<\/p>\n\n\n\n<p>Length of face diagonal of the cube $= 5.64$ units<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-practice-problems-on-diagonal-of-a-cube-formula\">Practice Problems on Diagonal of a Cube Formula<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Diagonal of a Cube Formula - Definition, Derivation, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">What will be the length of the face diagonal of a cube with a side of 12 inches?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">15 inches<\/div><div class=\"spq_answer_block\" data-value=\"1\">16.50 inches<\/div><div class=\"spq_answer_block\" data-value=\"2\">16.92 inches<\/div><div class=\"spq_answer_block\" data-value=\"3\">20.76 inches<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 16.92 inches<br\/>Length of face diagonal of the cube $= \\sqrt{2a} = \\sqrt{2} \\times 12 = 1.41 \\times 12 = 16.92$ inches<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">What will be the length of the body diagonal of a cube with a side of 6.1 inches?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">10 inches<\/div><div class=\"spq_answer_block\" data-value=\"1\">10.55 inches<\/div><div class=\"spq_answer_block\" data-value=\"2\">10.38 inches<\/div><div class=\"spq_answer_block\" data-value=\"3\">8.60 inches<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 10.55 inches<br\/>Length of body diagonal of the cube $= \\sqrt{3a} = \\sqrt{3} \\times 6.1 = 1.73 \\times 6.1 = 10.55$ inches<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Find the length of the side of a cube if the face diagonal is given as $9\\sqrt{2}$ inches.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">9 inches<\/div><div class=\"spq_answer_block\" data-value=\"1\">12.69 inches<\/div><div class=\"spq_answer_block\" data-value=\"2\">18 inches<\/div><div class=\"spq_answer_block\" data-value=\"3\">15.57 inches<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 9 inches<br\/>Length of face diagonal of the cube $= \\sqrt{2a} = 9\\sqrt{2}$<br>\r\nThus, $a = 9$ inches<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        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                  \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = \\\\sqrt{2} \\\\times 12 = 1.41 \\\\times 12 = 16.92$$ inches\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"15 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = \\\\sqrt{2} \\\\times 12 = 1.41 \\\\times 12 = 16.92$$ inches\"\n                                    }\n                                }, {\n                                \"@type\": 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   \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What will be the length of the body diagonal of a cube with a side of 6.1 inches?\",\n                    \"text\": \"What will be the length of the body diagonal of a cube with a side of 6.1 inches?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Length of body diagonal of the cube $$= \\\\sqrt{3a} = \\\\sqrt{3} \\\\times 6.1 = 1.73 \\\\times 6.1 = 10.55$$ inches\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"10 inches\",\n  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\\\\times 6.1 = 10.55$$ inches\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Length of body diagonal of the cube $$= \\\\sqrt{3a} = \\\\sqrt{3} \\\\times 6.1 = 1.73 \\\\times 6.1 = 10.55$$ inches\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Find the length of the side of a cube if the face diagonal is given as $$9\\\\sqrt{2}$$ inches.\",\n                    \"text\": \"Find the length of the side of a cube if the face diagonal is given as $$9\\\\sqrt{2}$$ inches.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = 9\\\\sqrt{2}$$<br>\r\nThus, $$a = 9$$ inches\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"12.69 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = 9\\\\sqrt{2}$$<br>\r\nThus, $$a = 9$$ inches\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"18 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = 9\\\\sqrt{2}$$<br>\r\nThus, $$a = 9$$ inches\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"15.57 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = 9\\\\sqrt{2}$$<br>\r\nThus, $$a = 9$$ inches\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"9 inches\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = 9\\\\sqrt{2}$$<br>\r\nThus, $$a = 9$$ inches\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Length of face diagonal of the cube $$= \\\\sqrt{2a} = 9\\\\sqrt{2}$$<br>\r\nThus, $$a = 9$$ inches\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-frequently-asked-questions-about-the-diagonal-of-a-cube-formula\">Frequently Asked Questions about the Diagonal of a Cube Formula<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-d5d7eca5-1096-448f-8c62-9e93c92a51b9\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-d5d7eca5-1096-448f-8c62-9e93c92a51b9\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d5d7eca5-1096-448f-8c62-9e93c92a51b9\"><strong>What is the diagonal of a cuboid?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-d5d7eca5-1096-448f-8c62-9e93c92a51b9\">\n\n<p>If a face diagonal lies on a face of a cuboid with dimensions of the face as a \u00d7 b, then its length is given by $\\sqrt{a^{2} + b^{2}}$.<\/p>\n\n\n\n<p>The body diagonal of a cuboid $= \\sqrt{l^{2} + b^{2} + h^{2}}$<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-d5d7eca5-1096-448f-8c62-9e93c92a51b9\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d5d7eca5-1096-448f-8c62-9e93c92a51b9\"><strong>What is the diagonal of a cube?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-d5d7eca5-1096-448f-8c62-9e93c92a51b9\">\n\n<p>The diagonal of a cube is a line segment connecting two opposite corners of the cube, passing through the center. It helps visualize the longest distance within the cube.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-d5d7eca5-1096-448f-8c62-9e93c92a51b9\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d5d7eca5-1096-448f-8c62-9e93c92a51b9\"><strong>What&#8217;s the formula for the face diagonal of a cube?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-d5d7eca5-1096-448f-8c62-9e93c92a51b9\">\n\n<p>The face diagonal of a cube is found using the formula: face diagonal = side length $\\times \\sqrt{2}$.\u00a0<\/p>\n\n\n\n<p>This helps calculate the diagonal within a face of the cube.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-d5d7eca5-1096-448f-8c62-9e93c92a51b9\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d5d7eca5-1096-448f-8c62-9e93c92a51b9\"><strong>How do you calculate the body diagonal of a cube?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-d5d7eca5-1096-448f-8c62-9e93c92a51b9\">\n\n<p>The body diagonal of a cube is given by the formula: body diagonal = side length $\\times \\sqrt{3}$.\u00a0<\/p>\n\n\n\n<p>This formula calculates the longest diagonal that passes through the cube&#8217;s body.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-d5d7eca5-1096-448f-8c62-9e93c92a51b9\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d5d7eca5-1096-448f-8c62-9e93c92a51b9\"><strong>How many diagonals are in a cube?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-d5d7eca5-1096-448f-8c62-9e93c92a51b9\">\n\n<p>There are 12 face diagonals and 4 body diagonals.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is the Diagonal of a Cube Formula? The formula to calculate the length of diagonal of a cube (body diagonal) is 3 times the length of one of its sides. In mathematical terms, if &#8220;s&#8221; represents the length of a side, then the diagonal &#8220;d&#8221; can be expressed as $d = s\\sqrt{3}$.\u00a0 A cube &#8230; <a title=\"Diagonal of a Cube Formula &#8211; Definition, Derivation, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/diagonal-of-a-cube-formula\" aria-label=\"More on Diagonal of a Cube Formula &#8211; Definition, Derivation, Examples\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-35389","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/35389","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=35389"}],"version-history":[{"count":8,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/35389\/revisions"}],"predecessor-version":[{"id":35408,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/35389\/revisions\/35408"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=35389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=35389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=35389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}