What is the volume of a 4D sphere?

Question

What is the volume of a 4D sphere? I've seen so many sites that would have answered this question, but all of them have so many numbers that some people (including me) don't understand. So I was hoping somebody here would give me a nice simple explanation.

Answer 1

Try with this explanation. Here the main result for a 4-dimensions sphere:

$$V_4(R) = \frac{1}{2} π^2 R^4$$

Answer 2

Let $V_n$ denote the volume of $S^n$. The projection $\pi(x_1, \dots, x_{n+1}) \to (x_1 \dots, x_n)$ maps each slice $\{(x_1, \dots, x_n, t)\in S^n\}$ onto $(1 - t^2)^{1/2} S^{n-1}$. It follows that \begin{align*} V_n &= V_{n-1} \int_{-1}^1 dt\;(1 - t^2)^{(n-1)/2}. \end{align*} The cases $n\leq 3$ are well-known, and the integral is not difficult to compute (or Mathematica, etc. will do it uncomplainingly).