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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 $⟨110⟩$ family, with the in-plane members satisfying $u+v+w=0$.

Nearest neighbors touch along a face diagonal: $4r=\sqrt{2}{a}_0$. HenceThe body diagonal is not the touching direction for this crystal structure.