For Poisson-Delaunay triangulations in $d$-dimensional Euclidean space $\mathbb{R}^{d}$, a structured and computationally efficient form of the probability density function (PDF) of the volume of a typical cell is analytically derived in this paper. In particular, the ensuing PDF and the corresponding cumulative density function (CDF) are exact and unified, applicable to spaces of arbitrary dimension ($d \ge 1$). Then, the special cases and shape characteristics of the resulting PDF are thoroughly examined. Finally, various applications of the obtained distribution functions are outlined and, in particular, a novel coordinated multi-point transmission scheme based on Poisson-Delaunay triangulation is developed and the pertinent void cell effect is precisely evaluated by using the obtained distribution functions.