When a nematic liquid crystal is confined to a capillary tube with strong homeotropic anchoring conditions, unstable +1 disclinations lines can branch into a pair of +1/2 lines forming loops attached to the +1 axial line through branch points. The shape of the +1/2 disclination lines is a function of the confinement and the Frank elasticity. Our previous work shows that nematic liquid crystals under cylindrical confinement display aradial-to-planar polar defect texture transition through the nucleation and uniform motion of a disclination branch point that separates a high charge disclination from two lower charge ones. Here, we present the existence of a branch point for a nematic LC confined to different conical geometries with homeotropic anchoring. Determination of the defect geometry in conjunction with our model provides a means of characterizing the elasticity of LCs. Our results show that a larger conical angle reduces the length at which disclination curvature relaxes to zero and it also leads to less bending energy. These new findings are useful to assess the Frank elasticity of the nematic LCs and predict novel defect structures under confinement.