{"id":9141,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9141"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"surface-area-and-volume-of-solids-conversion-of-solids-frustum-of-a-cone","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/surface-area-and-volume-of-solids-conversion-of-solids-frustum-of-a-cone\/","title":{"rendered":"Surface Area And Volume Of Solids, Conversion Of Solids, Frustum Of A Cone"},"content":{"rendered":"

Unit: <\/strong>Measurement System<\/strong><\/h2>\n

Chapter: <\/strong>Surface Area and Volume of Solids, Conversion of Solids, Frustum of a Cone<\/strong><\/h3>\n

Reference: – Introduction to Solids, Surface Area and Volume of Cuboid, Cube, Cylinder, Cone, Sphere, Conversion of Solids from One Shape to Another, Frustum of a Cone – Surface Area and Volume, Applications and Word Problems<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • The concepts of surface area and volume for common 3D shapes.<\/li>\n
  • How to calculate the surface area and volume of a frustum of a cone.<\/li>\n
  • The principle of conversion of solids and its applications.<\/li>\n
  • How to solve real-world problems involving these concepts.<\/li>\n<\/ul>\n

    Introduction to Solids<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    A solid is a three-dimensional object that has length, breadth, and height (or depth). Unlike 2D shapes, solids occupy space and have volume. Common solids include cubes, cuboids, cylinders, cones, and spheres.<\/p>\n

    The surface area is the total area of the outer surfaces of the solid, while the volume is the amount of space enclosed by the solid.<\/p>\n

    [Importance of Solids]<\/strong><\/p>\n

      \n
    • Essential for understanding objects in the real world, from boxes to buildings.<\/li>\n
    • Used in fields like architecture, engineering, and manufacturing.<\/li>\n
    • Helps in calculating material requirements, capacity, and cost.<\/li>\n
    • Forms the basis for more advanced topics in mathematics and physics.<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      A cardboard box<\/strong> is an example of a cuboid. Its surface area would be the area of cardboard used, and its volume would be the space inside it.<\/p>\n

      [Subtopics]<\/strong><\/p>\n

      1. Types of Solids<\/strong><\/p>\n

        \n
      • Polyhedra:<\/strong> Solids with flat faces (e.g., cube, cuboid, pyramid).<\/li>\n
      • Curved Solids:<\/strong> Solids with curved surfaces (e.g., cylinder, cone, sphere).<\/li>\n<\/ul>\n

        Key Points:<\/strong><\/p>\n

          \n
        • Lateral Surface Area (LSA):<\/strong> The area of all faces excluding the top and bottom.<\/li>\n
        • Total Surface Area (TSA):<\/strong> The area of all faces, including top and bottom.<\/li>\n
        • Volume:<\/strong> The measure of the space occupied by the solid.<\/li>\n<\/ul>\n

          Surface Area and Volume of Cuboid and Cube<\/strong><\/p>\n

          [Definition]<\/strong><\/p>\n

            \n
          • A cuboid<\/strong> is a solid with six rectangular faces. It has length (l), breadth (b), and height (h).<\/li>\n
          • A cube<\/strong> is a special cuboid where length = breadth = height = a.<\/li>\n<\/ul>\n

            [Importance of Cuboid and Cube]<\/strong><\/p>\n

              \n
            • Most common shapes for packaging and storage.<\/li>\n
            • Easy to model and calculate for various applications.<\/li>\n
            • Foundation for understanding more complex solids.<\/li>\n<\/ul>\n

              Examples<\/strong><\/p>\n

                \n
              • Find the TSA and volume of a cuboid with l=5 cm, b=4 cm, h=3 cm.<\/li>\n<\/ul>\n

                [Subtopics]<\/strong><\/p>\n

                1. Formulas for Cuboid<\/strong><\/p>\n

                  \n
                • Volume (V) = l × b × h<\/strong><\/li>\n
                • LSA = 2h(l + b)<\/strong><\/li>\n
                • TSA = 2(lb + bh + hl)<\/strong><\/li>\n<\/ul>\n

                  2. Formulas for Cube<\/strong><\/p>\n

                    \n
                  • Volume (V) = a³<\/strong><\/li>\n
                  • LSA = 4a²<\/strong><\/li>\n
                  • TSA = 6a²<\/strong><\/li>\n<\/ul>\n

                    Surface Area and Volume of Cylinder<\/strong><\/p>\n

                    [Definition]<\/strong><\/p>\n

                    A cylinder is a solid with two parallel circular bases connected by a curved surface. It has a height (h) and a base radius (r).<\/p>\n

                    [Importance of Cylinder]<\/strong><\/p>\n

                      \n
                    • Used in containers like cans, pipes, and tanks.<\/li>\n
                    • Common in mechanical and civil engineering.<\/li>\n
                    • Helps in understanding curved surface areas.<\/li>\n<\/ul>\n

                      Examples<\/strong><\/p>\n

                        \n
                      • Find the volume of a cylinder with r=7 cm and h=10 cm.<\/li>\n<\/ul>\n

                        [Subtopics]<\/strong><\/p>\n

                        1. Formulas for Cylinder<\/strong><\/p>\n

                          \n
                        • Volume (V) = πr²h<\/strong><\/li>\n
                        • Curved Surface Area (CSA) = 2πrh<\/strong><\/li>\n
                        • Total Surface Area (TSA) = 2πr(h + r)<\/strong><\/li>\n<\/ul>\n

                          Surface Area and Volume of Cone<\/strong><\/p>\n

                          [Definition]<\/strong><\/p>\n

                          A cone is a solid that tapers smoothly from a flat circular base to a point called the apex or vertex. It has a base radius (r), height (h), and slant height (l).<\/p>\n

                          [Importance of Cone]<\/strong><\/p>\n

                            \n
                          • Used in funnels, ice cream cones, and party hats.<\/li>\n
                          • Important in geometry and calculus.<\/li>\n
                          • Helps in understanding the concept of slant height.<\/li>\n<\/ul>\n

                            Examples<\/strong><\/p>\n

                              \n
                            • Find the slant height of a cone with r=3 cm and h=4 cm.<\/li>\n<\/ul>\n

                              [Subtopics]<\/strong><\/p>\n

                              1. Formulas for Cone<\/strong><\/p>\n

                                \n
                              • Slant Height (l) = √(r² + h²)<\/strong><\/li>\n
                              • Volume (V) = (1\/3)πr²h<\/strong><\/li>\n
                              • CSA = πrl<\/strong><\/li>\n
                              • TSA = πr(l + r)<\/strong><\/li>\n<\/ul>\n

                                Surface Area and Volume of Sphere<\/strong><\/p>\n

                                [Definition]<\/strong><\/p>\n

                                A sphere is a perfectly round geometrical object in three-dimensional space, like a ball. It is defined by its radius (r).<\/p>\n

                                [Importance of Sphere]<\/strong><\/p>\n

                                  \n
                                • Models objects like planets, balls, and bubbles.<\/li>\n
                                • Used in physics and astronomy.<\/li>\n
                                • Has the smallest surface area for a given volume.<\/li>\n<\/ul>\n

                                  Examples<\/strong><\/p>\n

                                    \n
                                  • Find the surface area of a sphere with r=7 cm.<\/li>\n<\/ul>\n

                                    [Subtopics]<\/strong><\/p>\n

                                    1. Formulas for Sphere<\/strong><\/p>\n

                                      \n
                                    • Volume (V) = (4\/3)πr³<\/strong><\/li>\n
                                    • Surface Area (SA) = 4πr²<\/strong><\/li>\n<\/ul>\n

                                      Conversion of Solids from One Shape to Another<\/strong><\/p>\n

                                      [Definition]<\/strong><\/p>\n

                                      This concept involves melting or reshaping a solid into another solid without any loss of material. The volume remains constant during conversion.<\/p>\n

                                      [Importance of Conversion]<\/strong><\/p>\n

                                        \n
                                      • Practical in metallurgy and manufacturing.<\/li>\n
                                      • Helps in solving problems involving material reuse.<\/li>\n
                                      • Tests understanding of volume conservation.<\/li>\n<\/ul>\n

                                        Examples<\/strong><\/p>\n

                                          \n
                                        • A metallic sphere of radius 6 cm is melted and recast into a cylinder of radius 3 cm. Find the height of the cylinder.<\/li>\n<\/ul>\n

                                          [Subtopics]<\/strong><\/p>\n

                                          1. Principle<\/strong><\/p>\n

                                          Volume of original solid = Volume of new solid<\/p>\n

                                          2. Application<\/strong><\/p>\n

                                          Set up an equation using the volume formulas of both solids and solve for the unknown dimension.<\/p>\n

                                          Frustum of a Cone<\/strong><\/p>\n

                                          [Definition]<\/strong><\/p>\n

                                          When a cone is cut by a plane parallel to its base, the portion between the base and the cutting plane is called a frustum of the cone. It has two circular bases of different radii.<\/p>\n

                                          [Importance of Frustum]<\/strong><\/p>\n

                                            \n
                                          • Common in buckets, lampshades, and certain architectural elements.<\/li>\n
                                          • Extends the understanding of cones to truncated shapes.<\/li>\n
                                          • Useful in practical volume and surface area calculations.<\/li>\n<\/ul>\n

                                            Examples<\/strong><\/p>\n

                                              \n
                                            • Find the volume of a frustum with radii 3 cm and 5 cm, and height 6 cm.<\/li>\n<\/ul>\n

                                              [Subtopics]<\/strong><\/p>\n

                                              1. Elements of a Frustum<\/strong><\/p>\n

                                                \n
                                              • R:<\/strong> Radius of the larger base.<\/li>\n
                                              • r:<\/strong> Radius of the smaller base.<\/li>\n
                                              • h:<\/strong> Height of the frustum (vertical distance between bases).<\/li>\n
                                              • l:<\/strong> Slant height of the frustum, \"\".<\/li>\n<\/ul>\n

                                                2. Formulas for Frustum<\/strong><\/p>\n

                                                  \n
                                                • Volume (V) = (1\/3)πh (R² + r² + Rr)<\/strong><\/li>\n
                                                • CSA = πl (R + r)<\/strong><\/li>\n
                                                • TSA = CSA + π(R² + r²)<\/strong><\/li>\n<\/ul>\n

                                                  Applications and Word Problems<\/strong><\/p>\n

                                                  [Definition]<\/strong><\/p>\n

                                                  These problems involve applying the formulas for surface area, volume, and conversion to real-life situations. They often require multiple steps and logical reasoning.<\/p>\n

                                                  [Importance of Word Problems]<\/strong><\/p>\n

                                                    \n
                                                  • Bridges theoretical math with practical application.<\/li>\n
                                                  • Enhances problem-solving and analytical skills.<\/li>\n
                                                  • Common in academic and competitive exams.<\/li>\n<\/ul>\n

                                                    Examples<\/strong><\/p>\n

                                                      \n
                                                    • A tent is in the shape of a cylinder surmounted by a cone. Find the canvas required for the tent.<\/li>\n<\/ul>\n

                                                      [Subtopics]<\/strong><\/p>\n

                                                      1. Problem-Solving Strategy<\/strong><\/p>\n

                                                        \n
                                                      1. Understand the problem and identify the solids involved.<\/li>\n
                                                      2. Note down the given dimensions.<\/li>\n
                                                      3. Determine which formulas are needed (SA, Volume, etc.).<\/li>\n
                                                      4. Perform the calculations step by step.<\/li>\n
                                                      5. Ensure units are consistent and interpret the result.<\/li>\n<\/ol>\n

                                                        [Example: -]<\/strong><\/p>\n

                                                        Problem Statement:<\/strong>
                                                        \nA solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones formed.
                                                        \nAlso, a bucket (frustum of a cone) has top and bottom radii of 28 cm and 21 cm respectively, and a height of 15 cm. Find its capacity in liters.<\/p>\n

                                                        Question:<\/strong> Solve both parts. Prove your answers by providing a step-by-step solution and giving three independent reasons<\/strong> supporting your conclusion for the first part from these domains: (A) Volume Conservation Principle, (B) Mathematical Calculation, (C) Logical Unit Analysis.<\/strong><\/p>\n

                                                        [Solution: -]<\/strong><\/p>\n

                                                        Part 1: Number of Cones Formed<\/strong><\/p>\n

                                                        Given:<\/strong><\/p>\n

                                                          \n
                                                        • Radius of sphere, \"\" cm<\/li>\n
                                                        • Radius of each cone, \"\" cm<\/li>\n
                                                        • Height of each cone, \"\" cm<\/li>\n<\/ul>\n

                                                          (A) Volume Conservation Principle<\/strong>
                                                          \nWhen a solid is melted and recast, its volume remains unchanged.
                                                          \nTherefore, Volume of Sphere = Number of cones × Volume of one cone.<\/p>\n

                                                          (B) Mathematical Calculation<\/strong>
                                                          \nFirst, calculate the volume of the sphere:
                                                          \n\"\"<\/p>\n

                                                          Compute \"\"<\/p>\n

                                                          \n\"\"<\/p>\n

                                                          Simplify step-by-step:
                                                          \nFirst, \"\"
                                                          \nThen, ..\"\"
                                                          \nLet's compute precisely:
                                                          \n\"\"Then, \"\" cm³.
                                                          \nSo, \"\" cm³.<\/p>\n

                                                          Now, volume of one cone:
                                                          \n\"\"<\/p>\n

                                                          Compute \"\"<\/p>\n

                                                          \n\"\"<\/p>\n

                                                          The 3 in numerator and denominator cancel:
                                                          \n\"\"
                                                          \n\"\", <\/em>\"\" cm³.<\/p>\n

                                                          Number of cones, \"\"<\/p>\n

                                                          Compute: \"\"
                                                          \nSo, \"\".<\/p>\n

                                                          (C) Logical Unit Analysis<\/strong>
                                                          \nThe volumes are both in cm³, so the ratio\"\" is a dimensionless number, correctly giving the number of cones. The calculation is consistent with unit analysis.<\/p>\n

                                                          Therefore, the number of cones formed is 126.<\/strong><\/p>\n

                                                          Part 2: Capacity of the Bucket (Frustum)<\/strong><\/p>\n

                                                          Given:<\/strong><\/p>\n

                                                            \n
                                                          • Top radius, R=28<\/em> cm<\/li>\n
                                                          • Bottom radius, r=21<\/em> cm<\/li>\n
                                                          • Height, h=15<\/em> cm<\/li>\n<\/ul>\n

                                                            The bucket is a frustum of a cone. Its capacity is its volume.
                                                            \nVolume of frustum:
                                                            \n\"\"<\/p>\n

                                                            Substitute the values:
                                                            \n\"\"<\/p>\n

                                                            Compute the terms inside:
                                                            \n\"\"
                                                            \n\"\"
                                                            \n\"\"Sum = 784+441+588=1813<\/em><\/p>\n

                                                            Now,
                                                            \n\"\"<\/p>\n

                                                            First, \"\"
                                                            \nSo, \"\"
                                                            \n\"\"39886<\/em>\/7=5698<\/em> cm³ (approximately, let's compute exactly).<\/p>\n

                                                            Actually, 1813<\/em>\/7=259<\/em> exactly? Let's check: 7×259=1813<\/em>. Yes!
                                                            \nSo, V=5×22×259=110×259=28490<\/em> cm³.<\/p>\n

                                                            Now, convert to liters. Since 1 liter = 1000 cm³,
                                                            \nCapacity = \"\" liters ≈ 28.5 liters<\/strong>.<\/p>\n

                                                            Final Answers:<\/strong><\/p>\n

                                                              \n
                                                            • Number of cones formed = 126<\/li>\n
                                                            • Capacity of the bucket = 28.5 liters (approximately)<\/li>\n<\/ul>\n

                                                              The solution for the number of cones is verified by the principle of volume conservation, precise mathematical computation, and logical unit analysis.<\/p>\n

                                                               <\/p>\n

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

                                                              Unit: Measurement System Chapter: Surface Area and Volume of Solids, Conversion of Solids, Frustum of a Cone Reference: – Introduction to Solids, Surface Area and Volume of Cuboid, Cube, Cylinder, Cone, Sphere, Conversion of Solids from One Shape to Another, Frustum of a Cone – Surface Area and Volume, Applications and Word Problems After studying […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[570],"tags":[],"class_list":["post-9141","post","type-post","status-publish","format-standard","hentry","category-math-sci-olympiad"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9141","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9141"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9141\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9141"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9141"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}