The volume and surface area of a ball/sphere in higher dimensions (also called the n-ball, hyperball, hypersphere) can found using the "gamma function," which a function that extends the factorial function to non-integers (so that something like 4.7! makes sense). The volume of the unit ball is highest in dimension 5. Why?
Click here for Part 2: https://www.youtube.com/watch?v=l4By6t3tyhs
Volume of a ball in 4-dimensional space.