Radius from Degree of Curve (Arc Definition)

Degree of curve is a nineteenth-century railroad invention that survives because it is easier to lay out in the field than a radius. Under the arc definition, D is the central angle that subtends exactly one 100 ft station measured along the arc, so R·D(in radians) = 100 ft and R = 100 × 180/π ÷ D = 5729.578/D feet. The crews building the transcontinental railroad carried printed tables of one-degree, two-degree and six-degree curves rather than radii, because a chainman could step off stations and turn deflection angles without ever knowing R. Railroads generally use the older chord definition, where the 100 ft is a chord rather than an arc; highways use the arc definition given here, and below about four degrees the two agree to within a foot.

The constant is unit-bound: 5729.578 is feet per degree, so the printed form works only in feet. This solver converts internally, so entering a radius in metres returns the same degree of curve. A worked example: a 4° curve has R = 5729.578/4 = 1,432.39 ft (436.6 m); a tight 12° urban curve is only 477 ft. Highway designers now specify R directly, but D still shows up on every railroad plan sheet and on older as-builts.