Euclidean ball

Volume

The n-dimensional volume of a Euclidean ball of radius r in n-dimensional Euclidean space is given by $${\displaystyle V_{n}(r)={\frac {\pi ^{\frac {n}{2}}}{\Gamma {\left({\frac {n}{2}}+1\right)}}}r^{n},}$$ where Γ is Leonhard Euler's gamma function (which can be thought of as an extension of the factorial function to fractional arguments). Using explicit formulas for particular values of the gamma function at the integers and half integers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function. These are: $${\displaystyle {\begin{aligned}V_{2k}(r)&={\frac {\pi ^{k}}{k!}}r^{2k}\,,\\[2pt]V_{2k+1}(r)&={\frac {2^{k+1}\pi ^{k}}{\left(2k+1\right)!!}}r^{2k+1}={\frac {2\left(k!\right)\left(4\pi \right)^{k}}{\left(2k+1\right)!}}r^{2k+1}\,.\end{aligned}}}$$