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Geometrical factor has been widely used in the design of induction tool and analysis of complex logging responses, as well as the high resolution processing of logging data. Works in this area are usually limited to two-dimensional isotropic medium and are only available to cases of coaxial transmitter and coaxial receiver coupling. For a more thorough understanding of multicomponent induction logging in the presence of transverse anisotropy, a knowledge of anisotropic geometrical factor is often necessary. In this paper, the two-dimensional and isotropic geometrical factors are extended to the anisotropic media using Born approximation, and the expressions of 3D Born geometrical factors for multicomponent induction logging are derived. Then the sensitivity and detectability of coaxial, coplanar and cross-coupling measurements are investigated. Numerical results show that with increasing coefficients of conductivity anisotropy, the spatial distribution of geometrical factors becomes increasingly complicated, and more sensitivity information can be detected by multicomponent induction tools. The multicomponent induction tool is sensitive to conductivity anisotropy at arbitrary dipping angles. In vertical wells, coplanar measurements are significantly affected by the conductivity anisotropy. Compared with coaxial and coplanar measurements, cross-coupling component offers superior sensitivity information to the conductivity anisotropy with the dipping angle being 40～ 60°. In horizontal wells, coaxial measurements are the most sensitive to the conductivity anisotropy. The extended 3D Born geometrical factor directly exhibits the anisotropy sensitivity in terms of spatial contribution, and has made up for the shortage of previous geometrical factors. The new geometrical factor will create favorable conditions for the development of new multicomponent induction tool and the interpretation of anisotropic formations.