Find the surface area of the n-1 sphere

Question

Q: Pick n>0 and suppose that Vol_n is the volume of the n-ball B(0,1), find the surface area of the n-1 sphere ∂B(ξ,ρ)?

I am totally stucked at where to start this problem. I have tried the transformation formula and the formula for finding surface areas itself. But neither of them seems to be related to the volume of the hypersphere. Need a hand! Thank you!

For more of you looking for answers for this problem, the link below is also useful..

https://www.quora.com/Geometry-Why-is-the-surface-area-of-a-sphere-the-derivative-of-the-volume-of-the-sphere

Answer 1

Hint: Increasing $\rho$ by a little bit, results in a change in volume which is proportional to the surface area (this is another way of saying that the derivative of the volume is the surface area)

Check that when $n=2,3$ , the familiar formulas do obey this rule: $\frac d{dr}(\pi r^2)=2\pi r$ and $\frac d{dr}... (\frac43\pi r^3)=4\pi r^2$.

So, we get

$\partial B(\xi,\rho)=\frac d{d\rho} (\operatorname {Vol_n} \rho^n)=n\cdot \operatorname {Vol_n}\cdot \rho^{n-1}$