We present a geometric method for approximating π using only chord lengths of regular polygons inscribed in a unit circle — specifically the equilateral triangle, square, pentagon, and hexagon — without any prior knowledge of π. The sum of the triangle and square chord lengths, √3 + √2 ≈ 3.14626, approximates π with relative error of approximately 0.15%. By incorporating chord lengths of the pentagon and hexagon, a refined formula is obtained: π ≈ √2 + √3 − √2/(φ·√3) + 1/2, accurate to within 5 × 10u207bu2075, where φ = (1+√5)/2 is the golden ratio. All components belong to the algebraic field Q(√2, √3, φ). A geometric constant Cu2080 = √2 + √3 − π ≈ 0.004671716 is identified and studied. It is shown that Cu2080 lies within the algebraic interval Au2082 < Cu2080 < B, where Au2082 = √2/(φ·√3) − 1/2 and B = (au2083 + au2085 − du2085 − au2086)·sin(60°), with bounds accurate to order 10u207bu2075. A trigonometric interpretation is established: Cu2080 = −cos(2θ), where θ = 45.134° represents a deviation of Cu2080/2 radians from the square's characteristic angle of 45°. An approximate relation to Euler's number, (π/4 − eu207b¹/u2074)/√2 ≈ Cu2080 with error below 7 × 10u207bu2076, is also noted. All numerical results are verified using 50-digit precision arithmetic.