 First two Brillouin zones of a rectangular lattice

The first Brillouin zone is $|k_x|<K,|k_y|<L$. The second consists of four disconnected lobes. Its right lobe has vertices $(K,-L),(2K,-L+K^2/L),(2K,L-K^2/L),(K,L)$; its upper lobe has vertices $(-K,L),(K,L),(0,L+K^2/L)$. Reflection in the coordinate axes gives the other two. The bisectors to $(2K,2L)$ and $(4K,0)$ truncate the lobes. Their combined area is $4KL$, equal to the first zone's area. Counting the Bragg planes crossed from the origin proves the construction.