DOI: https://doi.org/10.36347/sjpms.2025.v12i06.002

Pages: 208-216

The topic of this article means “one can construct a regular hexagon of which area is equal exactly to a given circle (O, r), using a straightedge and a compass only”. No scientific theory lasts forever, but specific research and discoveries continuously build upon each other. The three classic ancient Greek mathematical challenges likely referring to are “Doubling The Circle”, “Trisecting An Angle” and “Squaring The Circle”, all famously proven Impossible under strict compass-and-straightedge constraints, by Pierre Wantzel (1837) using field theory and algebraic methods [4], then also by Ferdinand von Lindemann (1882) after proving π is transcendental. These original Greek challenges remain impossible under classical rules since their proofs rely on deep algebraic/transcendental properties settled in the 19th century. Recent claims may involve reinterpretations or unrelated advances but do overturn these conclusions above. Among these, the "Squaring The Circle" problem and related problems involving π have captivated both professional and amateur mathematicians for millennia. The title of this paper refers to the concept of "constructing a regular hexagon that has the exact area of a given circle," or “Regular Hexagoning The Circle” for short. This research idea arose after the “Squaring The Circle” problem was studied and solved and published in “SJPMS” in 2024 [6].