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 = ??

http://mathworld.wolfram.com/Hypersphere.html // http://fr.scribd.com/doc/14789360/Le-probleme-de-lhyperchevre- — JJacquelin, Feb 16 '14 at 11:45
I didnt understand the concept of line element, thats why i asked here... thnx anyways. — abstractnature, Feb 16 '14 at 11:50
Possible duplicate of this question (see portion below equation (4)). — Andrew D. Hwang, Feb 16 '14 at 12:12

score 5 · Accepted Answer · answered Feb 16 '14 at 12:16

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.

score 1 · Answer 2 · edited May 13 '20 at 18:27

1

Another little known way to get this answer is to double integrate $\sqrt{1 + \nabla f \cdot \nabla f}$, where $f = \sqrt{R^2 - x^2 - y^2}$.

For hyper "areas" of geometric objects in $n$ dimensions, just integrate the above formula $n-1$ times, where $f = f (x_1, x_2, ...,x_{n-1})$.

edited May 13 '20 at 18:27

jeremy909

answered May 13 '20 at 17:38

Newman Red-pepper

2 Answers2