An important property of the Brillouin Zones is that, because the reciprocal lattice is periodic, there exists for any point outside the first zone a unique reciprocal lattice vector that will translate that point back inside the first zone. Each point in reciprocal space is only unique up to a reciprocal lattice vector. Each Zone contains every single physically distinguishable point, and so they all have the same area (in 2-D) or volume (in 3-D).
This is easiest to see by example. The illustrations below will show how the first six zones for the 2-D square and hexagonal lattices can be translated or 'folded' back on top of the first zone.
2-D square Zone folding
2-D hexagonal Zone folding