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In this scientific investigation, the article’s authors have created a three-dimensional computer simulation model to explore the sphericity and roundness of elements, specifically particles. The authors elucidated the detailed calculation of the sphericity of elements and roundness of elements for cylindrical particles through the application of the extreme-vertex 3D model. Based on the proposed extreme vertex model theory, the points at which linear constraints intersect were optimized, allowing for the maximum approximation of the form factor to unity. Furthermore, the proposed theory of extreme vertices in the context of modeling elements, particularly in voxel-based modeling, is of significant importance in defining the boundaries and limits of objects in three-dimensional space. Moreover, it was discovered that this idea holds importance in the context of the simplex-complex method, a commonly employed algorithm for addressing linear programming problems. Additionally, correlations between the sphericity and roundness of elements and the particle diameter were examined, enabling the prediction of particle shape. This is because the obtained sphericity values mostly approximate 1, which implies that the elements have a spherical or nearly spherical shape. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.