Abstract: By proving the equation of the volume of Wigner-Seitz primitive cell and the volume of primitive cell,we conclude that the volume of the first Brillouin zone is identical to the volume of reciprocal＇ s primitive cell. We establish an isometric hemeomorphism mapping between the n-th and （ n - 1 ）-th Brillouin zone to prove the equation of their volume.

Key words: Brillouin zone, isometric hemeomorphism mapping, inversion symmetry, Wigner- Seitz primitive cell