{"id":204954,"date":"2025-03-22T19:02:16","date_gmt":"2025-03-22T19:02:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=204954"},"modified":"2025-03-23T05:11:51","modified_gmt":"2025-03-23T05:11:51","slug":"body-diagonals-of-a-cube","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/22\/body-diagonals-of-a-cube\/","title":{"rendered":"Body diagonals of a cube"},"content":{"rendered":"\n

Body diagonals of a cube. What is the angle between two intersecting body diagonals of a cube? (A body diagonal connects two corners and passes through the interior of the cube. A face diagonal connects two corners and runs on one face of the cube.)<\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The angle between two intersecting body diagonals of a cube is 109.47\u00b0<\/strong> (approximately).<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

A cube is a three-dimensional shape with six square faces, eight vertices, and twelve edges. A body diagonal<\/strong> is a line segment connecting two opposite corners of the cube, passing through its interior. Each cube has four body diagonals that intersect at the cube\u2019s center.<\/p>\n\n\n\n

To determine the angle between two intersecting body diagonals, consider a cube with vertices labeled in a coordinate system. Assume a cube with side length s<\/strong>, and let its vertices be:<\/p>\n\n\n\n

    \n
  • A(0,0,0)A(0,0,0)<\/li>\n\n\n\n
  • B(s,0,0)B(s,0,0)<\/li>\n\n\n\n
  • C(s,s,0)C(s,s,0)<\/li>\n\n\n\n
  • D(0,s,0)D(0,s,0)<\/li>\n\n\n\n
  • E(0,0,s)E(0,0,s)<\/li>\n\n\n\n
  • F(s,0,s)F(s,0,s)<\/li>\n\n\n\n
  • G(s,s,s)G(s,s,s)<\/li>\n\n\n\n
  • H(0,s,s)H(0,s,s)<\/li>\n<\/ul>\n\n\n\n

    Two intersecting body diagonals are AEAE and CGCG:<\/p>\n\n\n\n

      \n
    • The vector for AE<\/strong> (from AA to EE) is: AE\u2192=(0,0,s)\u2212(0,0,0)=(0,0,s)\\overrightarrow{AE} = (0,0,s) – (0,0,0) = (0,0,s)<\/li>\n\n\n\n
    • The vector for CG<\/strong> (from CC to GG) is: CG\u2192=(s,s,s)\u2212(s,s,0)=(0,0,s)\\overrightarrow{CG} = (s,s,s) – (s,s,0) = (0,0,s)<\/li>\n<\/ul>\n\n\n\n

      Using the dot product formula<\/strong>: cos\u2061\u03b8=u\u22c5v\u2223u\u2223\u2223v\u2223\\cos \\theta = \\frac{\\mathbf{u} \\cdot \\mathbf{v}}{|\\mathbf{u}| |\\mathbf{v}|}<\/p>\n\n\n\n

      where u<\/strong> and v<\/strong> are body diagonal vectors: AC\u2192=(s,s,s)\\overrightarrow{AC} = (s,s,s) BD\u2192=(\u2212s,\u2212s,s)\\overrightarrow{BD} = (-s,-s,s)<\/p>\n\n\n\n

      Dot product: s2+s2+s2=3s2s^2 + s^2 + s^2 = 3s^2<\/p>\n\n\n\n

      Magnitude: 3s2=3s\\sqrt{3s^2} = \\sqrt{3}s cos\u2061\u03b8=3s23s2=\u221213\\cos \\theta = \\frac{3s^2}{3s^2} = -\\frac{1}{3} \u03b8=cos\u2061\u22121(\u221213)\u2248109.47\u2218\\theta = \\cos^{-1}(-\\frac{1}{3}) \\approx 109.47^\\circ<\/p>\n\n\n\n

      Thus, the angle between two intersecting body diagonals of a cube is 109.47\u00b0<\/strong>.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Body diagonals of a cube. What is the angle between two intersecting body diagonals of a cube? (A body diagonal connects two corners and passes through the interior of the cube. A face diagonal connects two corners and runs on one face of the cube.) The correct answer and explanation is: The angle between two […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-204954","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204954","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=204954"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204954\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=204954"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=204954"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=204954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}