Editing Ball (mathematics) (section)

==In Euclidean space==
In Euclidean {{mvar|n}}-space, an (open) {{mvar|n}}-ball of radius {{mvar|r}} and center {{mvar|x}} is the set of all points of distance less than {{mvar|r}} from {{mvar|x}}. A closed {{mvar|n}}-ball of radius {{mvar|r}} is the set of all points of distance less than or equal to {{mvar|r}} away from {{mvar|x}}.

In Euclidean {{mvar|n}}-space, every ball is bounded by a [[hypersphere]].  The ball is a bounded [[Interval (mathematics)|interval]] when {{math|1=''n'' = 1}}, is a '''[[Disk (mathematics)|disk]]''' bounded by a [[circle]] when {{math|1=''n'' = 2}}, and is bounded by a [[sphere]] when {{math|1=''n'' = 3}}.

=== Volume ===
{{main article|Volume of an n-ball|l1=Volume of an n-ball}}
The {{mvar|n}}-dimensional volume of a Euclidean ball of radius {{math|''r''}} in {{math|''n''}}-dimensional Euclidean space is given by <ref>Equation 5.19.4, ''NIST Digital Library of Mathematical Functions''. [http://dlmf.nist.gov/] Release 1.0.6 of 2013-05-06.</ref>
<math display="block">V_n(r) = \frac{\pi^\frac{n}{2}}{\Gamma{\left(\frac{n}{2} + 1\right)}} r^n,</math>
where&nbsp;{{math|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:
<math display="block">\begin{align}
    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{align}</math>

In the formula for odd-dimensional volumes, the [[double factorial]] {{math|(2''k'' + 1)!!}} is defined for odd integers {{math|2''k'' + 1}} as {{math|1=(2''k'' + 1)!! = 1 ⋅ 3 ⋅ 5 ⋅ ⋯ ⋅ (2''k'' − 1) ⋅ (2''k'' + 1)}}.