N-Dimensional Hyper-volume and Relationships between the Volumes of Hyper-spheres

Author(s)	Chung Seop Lee, Gabrielle Park, Ella Jang, Tony Oh, Lucas O, Ethan Choo, Carlson Bang, Hailey Yi, Alexander Chang, Trinity Jung
Country	United States
Abstract	The geometry of higher-dimensional Euclidean spaces extends familiar concepts such as length, area, and volume to arbitrary dimensions. In ℝⁿ, the closed n-dimensional ball of radius R is denoted by Bⁿ(R), while its boundary is the (n−1)-dimensional sphere Sⁿ⁻¹(R). This paper develops the concept of n-dimensional hypervolume, derives the general formula for the volume of Bⁿ(R), and examines relationships among volumes in different dimensions. Particular attention is given to recurrence relations, surface hyperarea, asymptotic behavior, and the striking fact that the volume of the unit n-ball tends to zero as n → ∞. Keywords: n-ball, hypersphere, hypervolume, Gamma function, Euclidean space, recurrence relation, high-dimensional geometry
Keywords	Topology, HyperSphere, Geometry, HyperVolume, 3-D Geometry, 3-D Sphere, 4-D Hyper Sphere, N-Dimensional Sphere
Field	Mathematics
Published In	Volume 7, Issue 5, September-October 2026
Published On	2026-09-07

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