Genuine multipartite entanglement (GME) represents the strongest form of quantum correlation, where all parties of a system across each bipartition are inseparably linked. This resource is crucial for advanced quantum protocols, including multi-party cryptography and distributed quantum computing, yet its quantification remains a significant theoretical challenge. This review surveys a rapidly evolving geometric paradigm that translates this challenge into a problem of polygon inequalities. By representing marginal bipartite entanglements as side lengths, the distribution of entanglement in a multipartite state must satisfy geometric closure conditions akin to forming a polygon. These polygon-based constraints provide a unifying framework for detecting and quantifying GME, offering an intuitive bridge between bipartite entanglement measures and global multipartite structure. We chart the development of this approach from its monogamy-based origins to modern formulations using concurrence triangles, wedge products, and entropy-based polygons. Recent advances have established rigorous conditions for these geometric quantities to serve as faithful, operationally meaningful entanglement measures, while also extending the framework to mixed states and higher-dimensional systems. Despite progress, key challenges persist in ensuring universal monotonicity and experimental robustness. By synthesizing these geometric insights, this review highlights how polygon inequalities are reshaping the quantification of complex quantum correlations and providing novel diagnostic tools for the next generation of quantum technologies.