Gateway Arch Arclength: Mathematical Derivation and Result

The Gateway Arch in St. Louis is a weighted catenary curve (not a parabola). Its shape is defined by:

y = A cosh(x/A) - A where A ≈ 68.767 m (225.6 ft), scaled to a 630 ft (192 m) span and height.

The arclength s of a catenary from x = -b to x = b is:

s = 2A sinh(b/A)

For the Arch, half-span b = 315 ft, A = 68.767 m ≈ 225.6 ft. Using the correct formula:

s = 2 × 225.6 × sinh(315 / 225.6) ≈ 451.2 × sinh(1.396) ≈ 451.2 × 1.896 ≈ 855 ft (261 m)

The previously cited formula s = a(e^x/y) and result of 1,151 ft are incorrect. The true structural length (centerline of the tube) is approximately 855–860 feet. The Arch's legs are 54 ft wide at the base, tapering to 17 ft at the top; the stainless steel skin adds negligible length.