Circular Arc Layout

Radius, arc length, and central angle of a circular arc from a measured chord (span) and rise (sagitta / middle ordinate) at midspan: R = (chord^2/4 + rise^2) / (2 x rise), central angle = 2 x acos((R - rise)/R), arc length = R x angle - the everyday layout question for an arch, curved trim, sheet-metal radius, or road curve (first-principles circle geometry).

Radius R = (chord^2 / 4 + rise^2) / (2 x rise); central angle = 2 x acos((R - rise) / R); arc length = R x angle. The rise is the sagitta (middle ordinate) at midspan.

Circular arc from a chord and rise (the sagitta / middle-ordinate relation) - first-principles circle geometry as in Machinery's Handbook (Industrial Press), by name; public domain.

Pure circle geometry, public; the chord and rise are user-supplied field measurements.

Estimate. AHJ and licensed professional govern.

Field names used by the API: chord_in, rise_in, radius_in, central_angle_deg, arc_length_in

• Chord and rise the chord (span) and the perpendicular rise at midspan are user-supplied measurements of the same arccircle geometry