A Geometric Approach to Computing π: From Inscribed Triangles to Precision Control

We present a formula for computing π that combines geometric intuition with direct precision control. Beginning with a triangle inscribed in a unit circle, we derive a simple computational formula using the small-angle approximation. We observe that this formula has a linear relationship between the exponent n and the number of correct decimal places obtained (2n + 3). By inverting this relationship, we obtain a refined formula where desired decimal precision d can be specified directly. The resulting formula computes π to exactly d correct decimal places for d ≥ 3, with predictable error scaling as 1.595 × 10^-(d+1). We derive this error bound using Taylor series analysis and validate the formula computationally to 110 decimal places.