Is first Brillouin zone the same as Wigner-Seitz cell?

[vbrasic]

Homework Statement

Not a homework question, but I am attempting to understand what exactly the first Brillouin zone is.

Homework Equations

The Attempt at a Solution

From my textbook, what I'm gathering is that one constructs the first Brillouin zone by constructing a "Wigner-Seitz" type cell in reciprocal space. My question is, is this how one constructs the first Brillouin zone, and if so, why/how does this work?

 

[vbrasic]

Essentially, my question is, suppose we have a hexagonal lattice, such that the reciprocal lattice vectors are given by, $$A=2\pi\hat{x}+\frac{2\pi}{\sqrt{3}}\hat{y},$$ and $$B=\frac{4\pi}{\sqrt{3}}\hat{y}.$$ The magnitudes of these are, $$|A|=|B|=\frac{4\pi}{\sqrt{3}}.$$ Is the Brillouin zone just the bound between, $$[-\frac{2\pi}{\sqrt{3}},\frac{2\pi}{\sqrt{3}}].$$ Because, as I understand it, because the magnitudes of reciprocal square lattice vectors are, $$\frac{2\pi}{a},$$ the Brillouin zone is essentially bisection of this (i.e. bisection of line to nearest neighbors as it is with Wigner-Seitz construction).