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ID: INPUT{id.yaml} Title: > Volume of the $d$-dimensional unit ball Definition: > $B_d = \text{vol}(\{x\in\mathbb{R}^d: |x|\leq 1\})$ is the volume of the unit ball in $d$-dimensional Euclidean space. Parameters: d: display: $d$ type: Z constraints: $d \geq 0$ Comments: Formulas: formula-gamma: > $B_d = \frac{\pi^{d/2}}{\Gamma(d/2+1)}$. formula-only-gamma: > $B_d = \frac{2^d\Gamma(1+1/2)^d}{\Gamma(1+d/2)}$. formula-sphere: > $dB_d = S_{d-1}$, where $S_{d-1}$ denotes the $(d-1)$-dimensional volume of the unit sphere in $\mathbb{R}^d$. Programs: References: Links: Wiki: title: "Wikipedia: Volume of an n-ball" url: https://en.wikipedia.org/wiki/Volume_of_an_n-ball Similar tables: Keywords: Tags: - elementary geometry - integral - period Data properties: type: R complete: no Display properties: number-header: $B_d$ Numbers: INPUT{numbers.yaml}
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Volume of the $d$-dimensional unit ball
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Numbers
$d$
$B_d$
INPUT{numbers.yaml} (not shown in preview)
Definition
$B_d = \text{vol}(\{x\in\mathbb{R}^d: |x|\leq 1\})$ is the volume of the unit ball in $d$-dimensional Euclidean space.
Parameters
$d$
— integer ($d \geq 0$)
Formulas
(1)
$B_d = \frac{\pi^{d/2}}{\Gamma(d/2+1)}$.
(2)
$B_d = \frac{2^d\Gamma(1+1/2)^d}{\Gamma(1+d/2)}$.
(3)
$dB_d = S_{d-1}$, where $S_{d-1}$ denotes the $(d-1)$-dimensional volume of the unit sphere in $\mathbb{R}^d$.
Links
[1]
Wikipedia: Volume of an n-ball
Data properties
Entries are of type: real number
Table is complete: no
