Surface area of a Hypersphere

Question

Hypersphere in 4 dimensions, I am having problem with finding the surface area of it. please help.

I know that surface area will have 3 dimensions in 4 dimensional space, I am having trouble to evaluate it

Circumference of circle= $2\pi r$

Surface area of sphere = $4\pi r^2$

Now, Surface area (or whatever, it will be having dimensions of a volume) of hypersphere = ??

Answer 1

The Wikipedia article gives the volume of the 4-dimensional hypersphere, with radius $r$, to be $$V = \frac{\pi^2}{2}r^4$$

The surface area can be found by differentiating with respect to $r$:

$$A = \frac{\mathrm{d}V}{\mathrm{d}r} = 2\pi^2r^3$$

The article also gives a proof of how to calculate the volume, and hence surface area.