Purpose <p>This study aims to investigate fundamental modal solutions for rectangular thin plates with four free edges: (1) to provide a novel criterion for fundamental mode identification, and (2) to derive high accurate explicit formulas for calculating the fundamental frequency.</p> Method <p>To achieve the study objectives, an investigation into the fundamental mode space was conducted, varying structural and material parameters including the plate’s dimensions (a, b), thickness (h), and Poisson’s ratio (ν). A novel, explicit criterion for fundamental mode identification was formulated, incorporating a six sigma tolerance tailored for specific aspect ratios. Ordinary least squares (OLS) multiple linear regression methodology was integrated within a dedicated process to identify optimal factor combinations for factorial frequencies based on 56 features. This methodology allowed for the derivation of factorial frequency formulas as a function of a, b, h, and ν, providing explicit expressions for calculating the fundamental frequency.</p> Conclusion <p>The study concludes that the global fundamental mode shapes of rectangular thin plates with four free edges are primarily governed by the aspect ratio a/b, Poisson’s ratio ν, and the thickness-induced shearing effect h on both dimensions. The derived explicit formulas for calculating the fundamental frequency exhibit exceptional accuracy, with a maximum error of −0.217% compared to extensive finite element analysis results, surpassing current state-of-the-art accuracy. This work presents physically meaningful, explicit formulations and introduces a novel perspective in plate and shell research, particularly relevant in scenarios with adjacent free edges and similar global modes.</p>