The following investigation is a product of the ongoing scientific inquiry ‘whence human suffering?‘, the same encountering a critical need to call into serious question the long-standing pi (π) “approximation” methodology (ie. of exhaustion) first employed by Archimedes (late, c. 287 – c. 212 BCE), and then by mathematicians and scientists ever since. To begin, the author draws attention to an important inquiry: ‘ does π ever naturally emerge as a product of a square? ‘ If so, it must be measureably so such to negate any/all need/inclining for “approximation” methodology(s) employing the use of multiple straight-edged polygons. Now consider the quadratic: x² – x – 1 = 0 and find it to have positive solution x = (1+√5)/2 which, as the reader may recognize, is the so-called golden ratio (hence: Φ). By expressing Φ in/on a base of 2π (thus generally applicable to rotational motion): Φ = (π+π√5)/2π = 1.618… and then squaring: Φ² = (3π+π√5)/2π = 2.618… we find a numerator difference (being a matter) of a discrete 2π: Φ² – Φ = 2π/2π …