N-Ball

#Math

$\displaystyle V_{n\text{-ball boundary}}=C_{n}r^{n-1},, C_{n}=n \frac{\pi^{n/2}}{\Gamma\left( \frac{n}{2}+1 \right)}$

• Desmos Graph

Derivation (in response to 3b1b Video):

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5. $\displaystyle \Gamma\left( \frac{n}{2}+1 \right)=\left( \frac{n}{2} \right)! \rightarrow C_{n}=n \frac{\pi^{n/2}}{\Gamma\left( \frac{n}{2} +1\right)}$

$\displaystyle V_{n\text{-ball}}=\frac{\pi^{n/2}}{\Gamma\left( \frac{n}{2}+1 \right)}r^{n}$

• Gamma Function