Let $x$ be the unit vector $(1,0,0,\ldots,0)$ in $\mathbb{R}^n$, and let $A(\theta)$ be the subset of $\mathcal{S}^{n-1}$ whose angle to $x$ is less than $\theta$, i.e. 
$$ A(\theta) = \left\{ y \in \mathbb{R}^n \; : \sum\_{k=1}^n y\_k^2 = 1 \; \text{and} \; y\_1 > \cos(\theta) \right\} . $$
I'd like to know the area of $A(\theta)$. The closest I've been able to find is [this question](https://mathoverflow.net/questions/102793/what-fraction-of-a-spheres-volume-lies-within-a-cone).


I can translate this into a question about multivariate Gaussians, and from thence get a double integral, but it's ugly, so I'm wondering if there's a simple formula.



What is the area of the piece of an $n$-sphere within a given angle of a vector?
This is 



> 
> [the volume of a ball in spherical geometry](http://en.wikipedia.org/wiki/Spherical_cap#Hyperspherical_cap)
> 
> 
> 


