Tungsten thin films are used for various applications and sometimes exhibit an A15 structure (b-W). They have some superior properties in comparison to the bcc structure (a-W), such as a higher superconducting transition temperature and larger spin Hall angle. However, elastic constants of b-W are unclear, which restricts mechanical applications and reliable density-functional-theory calculations. In this study, we synthesized a-W, b-W, and mixed-phase W films and determined their elastic constants by picosecond ultrasonics. We also calculated the elastic constants based on density functional theory and reveal that b-W has a larger elastic anisotropy and smaller shear modulus. Our calculation further indicates a stable stacking faulted b-W, which leads to a monoclinic structure.