Let $S_n(R)$ and $V_n(R)$ denote the surface area and volume of an $n$-sphere with radius $R$ respectively.
It is well known that $$S_n(R) = \frac{n \pi^{\frac{n}{2}}R^{n-1}}{\Gamma\left(\frac{n}{2}+1\right)}\quad\text{and}\quad V_n(R) = \frac{\pi^{\frac{n}{2}}R^{n}}{\Gamma\left(\frac{n}{2}+1\right)}$$ where $\Gamma(z)$ is the gamma function: $$\Gamma(z) = \int_0^\infty e^{-x} x^{z-1}\, dx.$$
Question: How to prove the two equations above?
I was given this question during an quantitative finance interview and asked to come out with the formulas.
I managed to recite the formulas but unable to prove them.
So I am looking for derivation which can be reproduced during an interview (short and concise proof would be better).