Step 1: Segment formula:
The line is at distance $d=1$ from the centre, and the circle has radius $r=2$. The half angle at the centre is $\alpha$ with $\cos\alpha=\tfrac12$, so $\alpha=\tfrac\pi3$.
Step 2: Sector minus triangle:
Sector angle $=2\alpha=\tfrac{2\pi}3$. Sector area $=\tfrac12r^2(2\alpha)=\tfrac12\cdot4\cdot\tfrac{2\pi}3=\tfrac{4\pi}3$.
Step 3: Triangle:
Triangle area $=\tfrac12r^2\sin2\alpha=\tfrac12\cdot4\cdot\tfrac{\sqrt3}2=\sqrt3$.
Step 4: Segment:
Segment area $=\tfrac{4\pi}3-\sqrt3$. Option (A).
Final Answer:
Sector minus triangle gives the same value.
\[ \boxed{A} \]