12. According to the instruction, the robot can complete the following actions on a plane: As shown in Figure 1, it first walks from the origin $O$ in a direction that is $\alpha\left(0 \leqslant \alpha \leqslant \frac{\pi}{2}\right)$ northeast for a period of time, then walks north for a period of time, but the time to change direction is uncertain. Assuming the robot's walking speed is $10 \mathrm{~m} / \mathrm{min}$, the possible area where the robot can be after walking for $2 \mathrm{~min}$ is $\qquad$
12. $(100 \pi-200) \mathrm{m}^{2}$.

As shown in Figure 1, let the position of the robot after walking for $2 \mathrm{~min}$ be point $P(x, y)$. Let the point where the robot changes direction be $A$, with $|O A|=a$, and $|A P|=b$. From the given conditions, we have $a+b=2 \times 10=20$, and
$$
\left\{\begin{array}{l}
x=a \cos \alpha, \\
y=a \sin \alpha+b .
\end{array}\right.
$$

Therefore, we have
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=a^{2}+2 a b \sin \alpha+b^{2} \leqslant(a+b)^{2}=400, \\
x+y=a(\sin \alpha+\cos \alpha)+b \geqslant a+b=20 .
\end{array}\right.
$$

Thus, the required plane figure is a segment, and its area is
$$
(100 \pi-200) \mathrm{m}^{2} \text {. }
$$