The main objective of this article is to develop a way for calculating the volume like element of an $ n $ -dimensional simplexin non-Euclidean spaces. The volume of a Euclidean tetrahedron can be calculated by this method,which encourages us to apply it in the case of non-Euclidean spaces. Although it can be useful to havean analytical solution to the problem but the solution will be more complicated as the number of dimensions increases.Compared to analytical methods, our methodology is some computational technique that is not only more practicalbut also very effective. First, we calculatethe volume like element for a non-Euclidean tetrahedron (3-simplex) and proceed to $ n $ dimensions,while specifying the volume like element in termsof the tetrahedron’s face angles, circumradii, and inradii.Also, we derive the formulas for the inradiiand circumradii of a spherical and hyperbolic triangle of use in calculating the volume like element.Moreover, we suggest a recursive approach to determining the volume like element in an $ n $ -dimensional non-Euclidean space,avoiding intricate computations in a higher-dimensional space.
