{"id":244485,"date":"2025-07-05T07:28:25","date_gmt":"2025-07-05T07:28:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244485"},"modified":"2025-07-05T07:28:27","modified_gmt":"2025-07-05T07:28:27","slug":"drag-and-drop-an-answer-to-each-box-to-correctly-complete-the-explanation-for-deriving-the-formula-for-the-volume-of-a-sphere","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/05\/drag-and-drop-an-answer-to-each-box-to-correctly-complete-the-explanation-for-deriving-the-formula-for-the-volume-of-a-sphere\/","title":{"rendered":"Drag and drop an answer to each box to correctly complete the explanation for deriving the formula for the volume of a sphere."},"content":{"rendered":"\n

Drag and drop an answer to each box to correctly complete the explanation for deriving the formula for the volume of a sphere. For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius r and height 2r minus the volume of two cones each with radius r and height r. A cross section of the sphere is a circle and a cross section of the cylinder minus the cones, taken parallel to the base of the cylinder, is an annulus. The volume of the cylinder with radius r and height 2r is 2\u00cf\u20acr^3. The volume of each cone with radius r and height r is (1\/3)\u00cf\u20acr^3. So the volume of the cylinder minus the two cones is (2\u00cf\u20acr^3) – 2((1\/3)\u00cf\u20acr^3). Therefore, the volume of the cylinder is (2\u00cf\u20acr^3) – (2\/3)\u00cf\u20acr^3, which simplifies to (4\/3)\u00cf\u20acr^3 by Cavalieri’s principle<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The volume of a sphere is derived by comparing it to a cylinder with two cones removed from it, using the concept of cross-sections and Cavalieri\u2019s principle.<\/p>\n\n\n\n

Step-by-Step Explanation:<\/h3>\n\n\n\n
    \n
  1. Cross Section of a Sphere<\/strong>:\n
      \n
    • A cross-section of a sphere, when sliced through any plane parallel to its base, produces a circle<\/strong>. The radius of this circle varies depending on the height at which the sphere is sliced.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • Cross Section of a Cylinder<\/strong>:\n
        \n
      • The cross-section of a cylinder with radius rrr and height 2r2r2r, when sliced parallel to its base, produces an annulus<\/strong> (a ring-shaped figure). The annulus represents the area between the outer boundary of the cylinder and the area inside the cross-section of the two cones.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Volume of the Cylinder<\/strong>:\n
          \n
        • The cylinder has a radius of rrr and height 2r2r2r. The formula for the volume of a cylinder is V=\u03c0r2hV = \\pi r^2 hV=\u03c0r2h, so the volume of the cylinder is: Volume\u00a0of\u00a0cylinder=2\u03c0r3\\text{Volume of cylinder} = 2\\pi r^3Volume\u00a0of\u00a0cylinder=2\u03c0r3<\/li>\n<\/ul>\n<\/li>\n\n\n\n
        • Volume of the Cones<\/strong>:\n
            \n
          • Two cones are removed from the cylinder. Each cone has a radius rrr and height rrr. The formula for the volume of a cone is: Vcone=13\u03c0r2hV_{\\text{cone}} = \\frac{1}{3} \\pi r^2 hVcone\u200b=31\u200b\u03c0r2h Substituting the height h=rh = rh=r, we get: Vcone=13\u03c0r3V_{\\text{cone}} = \\frac{1}{3} \\pi r^3Vcone\u200b=31\u200b\u03c0r3 Since there are two cones, the total volume of the two cones is: Total\u00a0volume\u00a0of\u00a0cones=2\u00d713\u03c0r3=23\u03c0r3\\text{Total volume of cones} = 2 \\times \\frac{1}{3} \\pi r^3 = \\frac{2}{3} \\pi r^3Total\u00a0volume\u00a0of\u00a0cones=2\u00d731\u200b\u03c0r3=32\u200b\u03c0r3<\/li>\n<\/ul>\n<\/li>\n\n\n\n
          • Subtracting the Volume of the Cones from the Cylinder<\/strong>:\n
              \n
            • The volume of the sphere is the volume of the cylinder minus the volume of the two cones: Volume\u00a0of\u00a0sphere=2\u03c0r3\u221223\u03c0r3\\text{Volume of sphere} = 2\\pi r^3 – \\frac{2}{3} \\pi r^3Volume\u00a0of\u00a0sphere=2\u03c0r3\u221232\u200b\u03c0r3 Simplifying this expression: Volume\u00a0of\u00a0sphere=(2\u03c0r3\u221223\u03c0r3)\\text{Volume of sphere} = \\left( 2\\pi r^3 – \\frac{2}{3} \\pi r^3 \\right)Volume\u00a0of\u00a0sphere=(2\u03c0r3\u221232\u200b\u03c0r3) =63\u03c0r3\u221223\u03c0r3= \\frac{6}{3} \\pi r^3 – \\frac{2}{3} \\pi r^3=36\u200b\u03c0r3\u221232\u200b\u03c0r3 =43\u03c0r3= \\frac{4}{3} \\pi r^3=34\u200b\u03c0r3<\/li>\n<\/ul>\n<\/li>\n\n\n\n
            • Final Formula<\/strong>:\n
                \n
              • Therefore, the volume of the sphere is: V=43\u03c0r3V = \\frac{4}{3} \\pi r^3V=34\u200b\u03c0r3<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

                This derivation uses Cavalieri’s principle, which states that two objects with the same cross-sectional area at every corresponding height must have the same volume. By comparing the sphere to a cylinder with cones removed, we can conclude that the volume of the sphere is 43\u03c0r3\\frac{4}{3} \\pi r^334\u200b\u03c0r3.<\/p>\n\n\n\n

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

                Drag and drop an answer to each box to correctly complete the explanation for deriving the formula for the volume of a sphere. For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with […]<\/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-244485","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244485","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=244485"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244485\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}