The radial application of the positive solution to (x² - x) = 1 as x = (√5 + 1) / 2

geometrically contains the area of the r = 1/2 circle: By simply setting the positive solution to (x² - x) = 1 as equal to 1/(πr²)²

one can directly solve for π without "approximating" it with n > 4 polygons: Use of the inverse square law completely

*circumvents*Archimedes' n > 4-sided polygon "approximation"

*deficiency*.

A 360° circle can only be reconciled by a/the figure whose internal angles sum the very same 360° (that is: a

*perfect square*).

To my knowledge, this is the first mathematical proof using such a law in agreement with observation that π ≠ 3.14159...

Let the area of a circle a be described as:

& let r = 1/2a = πr²

To calculate thea = π(1/2)²

a = π(1/4)

a = π/4

∴ π = 4a

*circumference*of the r = 1/2 circle,

use 1/(πr²)² = x:

To calculate the1/(πr²)² = x

1/(π(1/2)²)² = Φ

1/(π(1/4))² = Φ

1/(π/4)² = Φ

16/π² = Φ

π² = 16/Φ

π = 4/√Φ

∴ π = 2√(2(√5-1)))

≈ 3.1446055...

*area*of the r = 1/2 circle,

use 1/(4ar²)² = x:

The approximation of π using n > 4-sided polygons whose internal angles sum1/(4ar²)² = x

1/(4a(1/2)²)² = Φ

1/(4a(1/4))² = Φ

1/(4a/4)² = Φ

1/a² = Φ

a² = 1/Φ

a = 1/√Φ

∴ a = √(2(√5-1)))/2

≈ 0.78615137...

*greater*than 360° is the "

**" as:**

*Blunder of Millennia*only the polygon whose internal angles sum

*exactly*360° can be used to properly reconcile the circle.