While Smoothed Particle Hydrodynamics (SPH) is well-suited for simulating large deformations and material failure, its quantitative accuracy in solid mechanics is fundamentally limited by the inadequate geometric representation of particle volumes, especially near domain boundaries. The conventional practice of approximating initial particle volume as V_{0,i}\approx(\Delta x)^3introduces a critical discretization mismatch between the particle system and the actual continuum geometry, which systematically propagates errors into stress and internal force evaluations. To address this core issue, this study proposes a geometrically exact definition of the representative volume for each SPH particle, obtained by computing the precise intersection of its domain with the analytical solid geometry. This approach guarantees exact conservation of the total geometric volume in the initial configuration. Implemented within a reference-configuration-based corrected SPH framework, the method preserves this geometric consistency throughout the entire deformation process. Validation through three-dimensional benchmark simulations demonstrates that the proposed method achieves displacement and stress field predictions with accuracy comparable to established Finite Element Methods. The results conclusively show that an exact initial geometric representation of particle volumes is not merely a preprocessing refinement but an essential prerequisite for achieving quantitatively reliable SPH simulations in solid mechanics.