TITLE: Notation of sphere in $\Bbb{R}^n$.
QUESTION [7 upvotes]: In $\Bbb{R}^n$, the sphere centred at $x\in \Bbb{R}^n$ with radius $l$ is called $S^{n-1}(x,l)$ or an $(n-1)$-sphere. 
Why do we called it an $(n-1)$-sphere? It's certainly not defined on $\Bbb{R}^{n-1}$ instead of $\Bbb{R}^n$. 
Thanks in advance!

REPLY [6 votes]: Since the sphere is defined as the set of point on the sphere and not inside it, a sphere in $\mathbb{R}^n$ is just a $n-1$ dimensional surface. 
Another way of looking at this is that every point on the sphere with a given radius can be expressed by $n-1$ angles.