  Let $S$ denote the unit sphere of a real normed space. We show that the
intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$.
Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}\pi\|x-y\|,
\] where $d$ denotes the intrinsic metric on $S$.
