Formulating Mathematical Constants: Streamlining Shaded Area Calculations in Circle-Square Geometric Configurations

This research solves the difficulty in computing shaded areas of configurations of circles and squares using the derivation of two constants: ci=0.2146 for circles that inscribe and cc=0.5708 for circumscribed circles. The article makes these calculations easier by giving general shortcut equations that avoid the process steps traditionally associated with such problems. With verification against the standard techniques, the new constants provided exact and efficient solutions, considerably reducing the computational effort and the possibility of mistakes. The research contributes to geometry by filling a gap in the literature, namely that no streamlined approach previously existed for calculating shaded area. Practical applications of such findings are evident in school settings, mathematical competitions, and professional work in architecture and engineering. By simplifying these geometric computations, the study enhances accessibility and problem-solving efficiency. Beyond its immediate applications, the work shows the potential of these constants to inspire further developments in mathematics and related fields. Relevant to Euclidean geometry, optimization, and computational algorithms, the findings have interdisciplinary implications, extending to fields such as physics, design, and data visualization. This research establishes a foundation for future studies, including applications to irregular shapes and three-dimensional geometries and the development of computational tools to integrate these formulas into broader contexts.