The intrinsic metric on the unit sphere of a normed space

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 $x,y\in S$, $$\mathrm{\parallel }x-y\mathrm{\parallel }\le d(x,y)\le \sqrt{2}\pi \mathrm{\parallel }x-y\mathrm{\parallel },$$ where $d$ denotes the intrinsic metric on $S$.