A NOVEL AND ACCURATE METHOD FOR SQUARING THE CIRCLE WITH COMPASS ANS STRAIGHTEDGE

© Thomas J. O’ConnorIndependent Researcher Abstract This paper presents a novel and accurate geometric method forsquaring the circle using only compass and straightedge. Theconstruction does not require drawing a segment of length π,nor does it contradict the classical impossibility of constructingπ directly. Instead, the ratio π/2 arises naturally and indirectly from the geometry of a specific rectangle together with a quarter circle.A specific point determined solely by circle intersections, midpoints, and diagonal relationships is shown throughexact coordinate equations to lie precisely on the diagonal ofthe rectangle. This point fixes the height of the rectangle allowing the construction of a square whose area is exactlyequal to that of a given circle. The method is fully geometric,requires no approximations, and yields the highest accuracyachieved to date. It demonstrates an elegant relationshipbetween certain circular arcs and rectangular proportions.