A sphere with a radius of 7 cm is inscribed in a cube. What is the volume of the cube not occupied by the sphere?
["Title: How to Calculate the Unoccupied Volume in a Cube with an Inscribed Sphere (Radius 7 cm)", "Understanding the relationship between a sphere and the cube in which it is perfectly inscribed is a classic problem in geometry that reveals elegant mathematical principles. In this article, we explore what happens when a sphere with a radius of 7 cm fits snugly inside a cube—and how much volume remains unoccupied by the sphere.", "---", "### Understanding an Inscribed Sphere", "When a sphere is inscribed in a cube, it touches all six faces of the cube. The diameter of the sphere equals the length of the cube’s edge. Since the radius of the sphere is given as 7 cm, the diameter is:", "[\n\ ext{Diameter} = 2 \ imes 7,\ ext{cm} = 14,\ ext{cm}\n]", "Therefore, the edge length of the cube is also 14 cm.", "---", "### Volume of the Cube", "The volume ( V_{\ ext{cube}} ) of a cube with edge length ( s = 14,\ ext{cm} ) is:", "[\nV_{\ ext{cube}} = s^3 = 14^3 = 14 \ imes 14 \ imes 14 = 2744,\ ext{cm}^3\n]", "---", "### Volume of the Sphere", "The formula for the volume of a sphere is:", "[\nV_{\ ext{sphere}} = \frac{4}{3} \pi r^3\n]", "Substituting ( r = 7,\ ext{cm} ):", "[\nV_{\ ext{sphere}} = \frac{4}{3} \pi (7)^3 = \frac{4}{3} \pi (343) = \frac{1372}{3} \pi ,\ ext{cm}^3\n]", "Using ( \pi \approx 3.1416 ):", "[\nV_{\ ext{sphere}} \approx \frac{1372}{3} \ imes 3.1416 \approx 457.33 \ imes 3.1416 \approx 1436.76,\ ext{cm}^3\n]", "---", "### Unoccupied Volume in the Cube", "The volume not occupied by the sphere is the cube’s volume minus the sphere’s volume:", "[\nV_{\ ext{unoccupied}} = V_{\ ext{cube}} - V_{\ ext{sphere}} = 2744 - \frac{1372}{3} \pi\n]", "For a simpler, exact answer:", "[\nV_{\ ext{unoccupied}} = 2744 - \frac{1372}{3} \pi ,\ ext{cm}^3\n]", "For a numerical approximation:", "[\nV_{\ ext{unoccupied}} \approx 2744 - 1436.76 = 1307.24,\ ext{cm}^3\n]", "---", "### Conclusion", "A sphere with a radius of 7 cm inscribed in a cube has a volume of approximately 1436.76 cm³, while the cube holds 2744 cm³. The difference—about 1307.24 cm³—represents the volume of the cube not occupied by the sphere. This calculation supports key concepts in geometry, helping visualize space relationships that are vital in architecture, engineering, and design.", "---", "### Key Takeaways:\n- Cube edge length = diameter of inscribed sphere = 14 cm\n- Cube volume = 2744 cm³\n- Sphere volume = ( \frac{1372}{3} \pi ) cm³ ≈ 1436.76 cm³\n- Unoccupied volume ≈ 1307.24 cm³", "Understanding these relationships strengthens spatial reasoning and mathematical intuition—essential for students, educators, and professionals alike.", "---", "Keywords for SEO: sphere inscribed in cube, volume of cube not occupied by sphere, cube and sphere geometry, 7 cm sphere volume, unoccupied space cube 14 cm edge, geometric problem solution, cube volume formula."]
