### Instruction: A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z} \ | \ z \in R\right\rbrace$.  Find the area of $S.$.### Response: I need to find the area of the region that is the inverse of the region outside the hexagon.
The inverse of a point $z = x + iy$ in the complex plane is $\frac{1}{z} = \frac{x}{x^2 + y^2} - i\frac{y}{x^2 + y^2}$, which is also the point that is symmetric to $z$ with respect to the unit circle.
So, the inverse of the region outside the hexagon is the region inside a smaller hexagon that is symmetric to the original one with respect to the unit circle.
To find the area of this smaller hexagon, I need to find the length of its sides, which is the inverse of the distance between opposite sides of the original hexagon.
The distance between opposite sides of the original hexagon is one unit, so the length of the sides of the smaller hexagon is $\frac{1}{1} = 1$ unit.
The area of a regular hexagon with side length $s$ is $6 \times \frac{s^2 \sqrt{3}}{4} = \frac{3s^2 \sqrt{3}}{2}$, so the area of the smaller hexagon is $\frac{3 \sqrt{3}}{2}$ square units.
Therefore, the area of $S$ is the area of the smaller hexagon.