We propose a numerical method that enables the calculation of volume fractions from triangulated surfaces immersed in unstructured meshes. First, the signed distances are calculated geometrically near the triangulated surface. For this purpose, the computational complexity has been reduced by using an octree space subdivision. Second, an approximate solution of the Laplace equation is used to propagate the inside/outside information from the surface into the solution domain. Finally, volume fractions are computed from the signed distances in the vicinity of the surface. The volume fraction calculation utilizes either geometrical intersections or a polynomial approximation based on signed distances. An adaptive tetrahedral decomposition of polyhedral cells ensures a high absolute accuracy. The proposed method extends the admissible shape of the fluid interface (surface) to triangulated surfaces that can be open or closed, disjoint, and model objects of technical geometrical complexity.Current results demonstrate the effectiveness of the proposed algorithm for two-phase flow simulations of wetting phenomena, but the algorithm has broad applicability. For example, the calculation of volume fractions is crucial for achieving numerically stable simulations of surface tension-driven two-phase flows with the unstructured Volume-of-Fluid method. The method is applicable as a discrete phase-indicator model for the unstructured hybrid Level Set/Front Tracking method.The implementation is available on GitLab [27]. Program summaryProgram Title: argo/triSurfaceImmersionCPC Library link to program files:https://doi.org/10.17632/7g72xrcgjp.1Developer's repository link:https://gitlab.com/leia-methods/argoLicensing provisions: GPLv3Programming language: C++Nature of problem: Computing volume fractions and signed distances from triangulated surfaces immersed in unstructured meshes.Solution method: First, the algorithm computes minimal signed distances between mesh points (cell centers and cell corner-points) and the triangulated surface, in the close vicinity of the surface. The sign is computed with respect to the surface normal orientation. Afterwards, the sign is propagated throughout the unstructured volume mesh by an approximate solution of a diffusion equation. The bulk cells' volume fractions are set, and interface cells are identified based on the signed distances. Volume fractions in cells intersected by the triangulated surface mesh are either computed by geometric intersections between surface triangles and a cell or by an approximation of the volume fraction approximation from signed distances, coupled with tetrahedral cell decomposition and refinement.Additional comments including restrictions and unusual features: The volume mesh can consist of cells of arbitrary shape. The surface mesh normal vectors need to be oriented consistently.