Circular Segment Area (from Chord and Rise)

The piece the circle-layout family gives lengths and angles for but never the area. From the same chord and rise (sagitta): R = (chord^2/4 + rise^2)/(2 x rise), central angle theta = 2 acos((R - rise)/R), segment area A = (1/2) R^2 (theta - sin theta). A 24 in chord bowed 4 in is a 20 in radius, a 73.74-deg arc, and a 65.40 in^2 segment; a chord equal to the diameter gives exactly half the circle. The arched-panel blank, the partly-filled round-tank/pipe cross-section, or the waste cut to make an arched opening. Area only, not arc length; first-principles geometry as in Machinery's Handbook.

R = (chord^2/4 + rise^2)/(2 x rise); theta = 2 x acos((R - rise)/R); segment area A = (1/2) R^2 (theta - sin theta) = R^2 acos((R-h)/R) - (R-h) sqrt(2 R h - h^2).

Area of a circular segment (the region between a chord and its arc) - 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 of the same arc.

Estimate. AHJ and licensed professional govern.

Field names used by the API: chord_in, rise_in, segment_area_in2, radius_in, central_angle_deg

• Segment area A = (1/2) R^2 (theta - sin theta), the sector minus the trianglecircle geometry / Machinery's Handbook
• Chord and rise R and theta from the chord (span) and rise (sagitta) of the same arccircle geometry