= Solution
A <face-centered cubic> cell has atoms at its corners and face centers. The $(111)$ plane intercepts the three positive cubic axes at $(a_0,0,0)$, $(0,a_0,0)$ and $(0,0,a_0)$; it is a close-packed triangular layer. Close-packed directions are the $\langle110\rangle$ family, with the in-plane members satisfying $u+v+w=0$.
Nearest neighbors touch along a face diagonal: $4r=\sqrt2a_0$. Hence
$$
\boxed{r=\frac{a_0}{2\sqrt2}=1.389\,\mathrm{\mathring A}}.
$$
The body diagonal is not the touching direction for this crystal structure.
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