Solution: The volume of a sphere with radius $ 2r $ is:

["Title: How to Calculate the Volume of a Sphere with Radius 2r: A Clear Solution", "Understanding the volume of a sphere is essential in mathematics, physics, engineering, and everyday problem-solving. If you're asking, “What is the volume of a sphere with radius 2r?”—this article provides a clear, step-by-step solution to compute the volume, explain the formula, and highlight why this knowledge matters.", "---", "### The Volume of a Sphere Formula", "The volume ( V ) of a sphere is calculated using the well-known mathematical formula:", "[\nV = \dfrac{4}{3} \pi r^3\n]", "Here:\n- ( V ) = volume of the sphere\n- ( r ) = radius of the sphere\n- ( \pi ) ≈ 3.14159 (pi), a mathematical constant approximately equal to 3.14", "This formula describes how much three-dimensional space a sphere occupies. For a sphere of radius ( r ), the volume grows cubically with the radius — meaning even small increases in radius significantly expand the volume.", "---", "### Computing Volume for Radius 2r", "When the sphere’s radius is ( 2r ), substitute ( 2r ) into the volume formula:", "[\nV = \dfrac{4}{3} \pi (2r)^3\n]", "First, compute ( (2r)^3 ):", "[\n(2r)^3 = 8r^3\n]", "Now plug this back:", "[\nV = \dfrac{4}{3} \pi (8r^3) = \dfrac{32}{3} \pi r^3\n]", "So, the volume of a sphere with radius ( 2r ) is:", "[\n\boxed{ \dfrac{32}{3} \pi r^3 }\n]", "---", "### Why This Matters", "This calculation is not just theoretical. Knowing the volume with radius ( 2r ) helps in real-world applications such as:", "- Engineering design, where knowing how much material or space a spherical tank or component occupies is vital.\n- Physics and astronomy, where objects like planets (approximated as spheres) are analyzed using volume-based mass and density calculations.\n- Geometry education, where mastering volume formulas builds a strong foundation for 3D spatial reasoning.", "---", "### Comparing Volumes: Radius ( r ) vs. ( 2r )", "A simple way to see the relationship is comparing volumes at the same radius:", "- Volume with radius ( r ):\n[\nV_1 = \dfrac{4}{3} \pi r^3\n]", "- Volume with radius ( 2r ):\n[\nV_2 = \dfrac{32}{3} \pi r^3\n]", "Clearly, ( V_2 = 8 \ imes V_1 ), because doubling the radius increases volume by a factor of eight — consistent with the cubic dependence on radius.", "---", "### Final Thoughts", "Mastering volume calculations empowers precise measurements and deep understanding of spatial properties. Whether you're studying math, building models, or solving practical problems, knowing the volume of a sphere with radius ( 2r ) gives you a clear, efficient method to determine its space-occupying capacity.", "(\blacksquare)", "---", "Keywords: Volume of sphere, sphere volume formula, radius 2r, math solution, geometry, cubic formula, pi, spatial reasoning, educational resource, engineering application, planetary volume, mathematical derivation", "---", "Meta Description:\nLearn how to calculate the volume of a sphere with radius 2r using the formula ( V = \dfrac{32}{3} \pi r^3 ). This step-by-step solution explains the math and real-world relevance for easy understanding."]