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The closed form for the volume of the unit $n$ -ball

Statement

For every $n \geq 1$ , $V_{n} (1) = π^{n / 2} / Γ (n / 2 + 1)$ . Here $n$ is an integer.

Facts & Assumptions

Given: Unit-ball volumes $V_{n} (1)$ in positive integer dimensions.

[F1]

One has $V_{1} (r) = 2 r$ for $r \geq 0$ , and for $n \geq 2$ , $V_{n} (1) = V_{n - 1} (1) \int_{- 1}^{1} (1 - t^{2})^{(n - 1) / 2} d t$ (Closed Euclidean balls are Jordan measurable and their volumes satisfy the slicing recursion).

[F2]

For $p, q > 0$ , $B (p, q) = Γ (p) Γ (q) / Γ (p + q)$ (The real Beta--Gamma identity).

[F3]

$Γ (1 / 2) = \sqrt{π}$ ( $Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral).

[F4]

For $s > 0$ , $Γ (s + 1) = s Γ (s)$ (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

Proof

technique · induction

1.1F1F3F4basealgebra

In dimension $1$ , [F1] gives $V_{1} (1) = 2$ , while [F3] and [F4] give $π^{1 / 2} / Γ (3 / 2) = \sqrt{π} / ((1 / 2) \sqrt{π}) = 2$ .

1.2ihF1F2F3algebra

Assume the formula in dimension $n - 1$ . In the integral of [F1], symmetry followed by $u = t^{2}$ gives $\int_{- 1}^{1} (1 - t^{2})^{(n - 1) / 2} d t = B (1 / 2, (n + 1) / 2)$ . By [F2] and [F3], this equals $\sqrt{π} Γ ((n + 1) / 2) / Γ (n / 2 + 1)$ .

2.1step 1.2discharge-inductionalgebra∎

Multiply the expression in step 1.2 by the induction value $π^{(n - 1) / 2} / Γ ((n + 1) / 2)$ ; the common Gamma factor cancels, giving $V_{n} (1) = π^{n / 2} / Γ (n / 2 + 1)$ . This discharges the induction.

Depends on

Used by

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2.4(d) (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, §5C (standard reference, not scraped)

The closed form for the volume of the unit nn-ball