= Solution
The stipulated conventional cell is a square of side $2a$. Although introduced as a <centered rectangular lattice>, this equal-side case is a <square Bravais lattice> viewed with a rotated primitive basis. A corner-to-center separation is
$$
d_{\rm nn}=\sqrt{a^2+a^2}=\sqrt2\,a,
$$
which is shorter than the $2a$ separation between adjacent conventional corners. Consequently the first contact of equal circles occurs at
$$
2r=d_{\rm nn},\qquad r=\frac{a}{\sqrt2}.
$$
The conventional square represents two <lattice points>: one from its four shared corners and one from its center. Its <packing fraction> is therefore
$$
\boxed{\eta=\frac{2\pi r^2}{(2a)^2}=\frac{\pi}{4}\simeq0.7854}.
$$
Equivalently, one circle occupies each primitive area $2a^2$. This is square <circle packing>; the drawing does not define a triangular close-packed lattice.
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