A Geometric Approach to Computing π: From an Inscribed Triangle to Precision Control

This paper describes a geometric method for computing π using a right triangle inscribed in a unit circle. For small angles, the triangle's base approximates the sector's arc length, which gives a simple formula: f(n) = 180 · 10^n sin(π/(180 · 10^n)). For n = 0 to n = 13, this formula delivers exactly 2n+3 correct decimal places. That's 3 to 29 decimals, more than enough for any calculator or standard floating-point arithmetic. Beyond this range, edge cases occasionally yield one fewer decimal when specific digits of π are 0 or 1. Taylor series analysis showed me the cubic term was the problem. I derive a correction factor of (3/20)θ³ that eliminates these edge cases, guaranteeing a minimum of 2n+1 decimal places with 100% reliability while typically achieving 2n+3 to 2n+4 decimals. The refined formula has been validated for n up to 1000 (over 2000 decimal places). I also develop a parametric form where you can directly specify how many decimals you want through the relationship α(d) = (d-1)/2.