We propose and theoretically study a new hollow-core fiber-based curvature sensing approach with the capability of detecting both curvature radius and angle. The new sensing method relies on a tubular-lattice fiber that encompasses, in its microstructure, tubes with three different thicknesses. By adequately choosing the placement of the tubes within the fiber cross-section, and by exploring the spectral shifts of the fiber transmitted spectrum due to the curvature-induced mode field distributions’ displacements, we demonstrate a multi-axis curvature sensing method. In the proposed platform, curvature radii and angles are retrieved via a suitable calibration routine, which is based on conveniently adjusting empirical functions to the fiber response. Evaluation of the sensing method performance for selected cases allowed the curvature radii and angles to be determined with percentual errors of less than 7%. The approach proposed herein provides a promising path for the accomplishment of new curvature sensors able to resolve both the curvature radius and angle.