The 2:1 Ratio Hiding in π

This paper extends my previous geometric formula for π [1] by combining two natural bounds on the arc of a circle. My previous formula [1] approximated the arc with the chord (length sin θ), which underestimates it. This paper adds the tangent line (length tan θ), which overestimates it. If I combine these two approximations in a 2:1 ratio, the cubic terms in the Taylor series cancel exactly. The leading error drops from θ³ to θ⁵. The formula is f(n) = 180 · 10ⁿ · [2 sin(θ) + tan(θ)]/3, where θ = π/(180 · 10ⁿ). It delivers 4n+7 correct decimal places with a guaranteed minimum of 4n+3. I tested it to 20 million digits without encountering an edge case. Version 2 note: Reframed geometric motivation in terms of chord and tangent line approximations to the arc. The formula is unchanged. Extended testing from 2 million to 20 million digits. Updated figures.