The Cartesian product of two measure spaces has a natural measure. So for a fibre bundle whose base and fiber spaces are measure spaces. As the volume of the product space is the product of the volumes of the factor spaces, so is the volume of a volume of fiber bundle space the product of the volumes of the base and fiber spaces.

The special orthogonal Lie group, SO_{n}(R) is a fibre bundle whose base space is S(n−1) and whose fibre is SO_{n−1}(R).
Ergo the volume of SO_{n−1}(R) is ∏nγ_{n} for r ranging from 1 to n and where γ_{n} is the volume of the unit n ball.

γ_{n} = π^{n/2}/((n/2)!)

0!=1

(−½)! = √π;

x! = x((x−1)!)