= Solution
The <first Brillouin zone> is the origin-centered <Wigner-Seitz cell> of the <reciprocal lattice>. Its nearest reciprocal points are $\pm\mathbf b_1$ and $\pm\mathbf b_2$. The perpendicular bisectors follow from
$$
|\mathbf k|^2=|\mathbf k-\mathbf G|^2
\quad\Longleftrightarrow\quad
\mathbf k\cdot\mathbf G=\frac{|\mathbf G|^2}{2}.
$$
Consequently the zone is
$$
\boxed{|k_x+k_y|\leq\frac{\pi}{a},\qquad
|k_y-k_x|\leq\frac{\pi}{a}}.
$$
It is the diamond with vertices
$$
\boxed{(\pi/a,0),\ (0,\pi/a),\ (-\pi/a,0),\ (0,-\pi/a)}.
$$
The reciprocal-space sketch above shows these boundaries. Its area is $2\pi^2/a^2=(2\pi)^2/A_c$, where $A_c=2a^2$. This area check rules out mistakenly using the nonprimitive conventional cell to define a smaller first zone.
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