It does indeed scale with r^(n-1), but your factors are not close at all. It involves the gamma function, which in this case can be expanded into various factorials and also a factor of sqrt(pi) when n is odd. According to Wikipedia, the expression is 2pi(n/2)r(n-1)/Gamma(n/2).
Edit: ignore the below, I forgot my pi-factor in the gamma function for half-integers…
Edit 2: Since you’re right, my missing gamma-factor completely changes this. An infinite-dimensional hypersphere will have zero surface area for any (finite?) radius.
Original dum-dum:
While I’m completely open that my factor is likely wrong here, the expression you provided is definitely wrong in the 3D case (I’m assuming the r superscript on the pi was a typo), since it doesn’t give n = 3 => A = 4 pi r^2.
It does indeed scale with r^(n-1), but your factors are not close at all. It involves the gamma function, which in this case can be expanded into various factorials and also a factor of sqrt(pi) when n is odd. According to Wikipedia, the expression is 2pi(n/2)r(n-1)/Gamma(n/2).
Edit: ignore the below, I forgot my pi-factor in the gamma function for half-integers…
Edit 2: Since you’re right, my missing gamma-factor completely changes this. An infinite-dimensional hypersphere will have zero surface area for any (finite?) radius.
Original dum-dum:
While I’m completely open that my factor is likely wrong here, the expression you provided is definitely wrong in the 3D case (I’m assuming the r superscript on the pi was a typo), since it doesn’t give n = 3 => A = 4 pi r^2.