High-D Sphere Volume: Peaks at n=5, Then Shrinks to Zero

Unit sphere volume peaks at n=5 (V₅ ≈ 5.264), then collapses toward zero. In high dimensions, the sphere is negligible inside its own bounding box.

V_n (unit sphere volume)
Log-Scale V_n vs n
Key Values
nV_n% of box
At n = 5
V = 5.2638
Fraction of bounding box: 16.45%
Sphere is 16.45% of its bounding box