segmentarea

Segmentarea refers to the area of a circular segment, the region of a circle that lies between a chord and the arc that subtends that chord. The segment is defined by the circle’s radius and the central angle θ (in radians) subtending the chord.

If the circle has radius r and the segment corresponds to a central angle θ, its area is A = (r^2/2) (θ − sin θ). This formula arises from subtracting the area of the isosceles triangle formed by the two radii and the chord from the area of the corresponding sector. For minor segments, 0 < θ < π; for major segments, π < θ < 2π, and the same expression can be applied using the appropriate θ or by using the complementary angle 2π − θ to obtain the larger area.

Alternative expressions exist when different measurements are given. If the segment height h (the distance from the chord to the arc) is known, the area can be computed as A = r^2 arccos((r − h)/r) − (r − h) sqrt(2 r h − h^2). If the chord length c is known, r and θ can be related to c, and A can be derived accordingly.

Segmentarea has applications in geometry, surveying, CAD and design, optics, and any context requiring precise calculations of circular portions. It is closely related to the concepts of circle area, sector area, and triangle area, and is often discussed in discussions of circular geometry and integration.