kreissegment

Kreissegment, in geometry, is the region of a circle bounded by a chord and the arc that subtends that chord. The term corresponds to the English phrase circle segment. If the arc is the smaller one (less than 180 degrees) it is called a minor Kreissegment; if the arc is larger than a semicircle, it is a major Kreissegment.

Key quantities associated with a Kreissegment are the circle’s radius R, the chord length c, the sagitta h (the distance from the chord to the arc), and the central angle theta in radians subtending the arc. The sagitta and chord relate to the radius by h = R − sqrt(R^2 − (c/2)^2). The central angle relates to the chord by theta = 2 arcsin(c/(2R)) = 2 arccos((R − h)/R).

The area of a Kreissegment can be expressed in two common forms. In terms of the central angle, A = (R^2/2)(theta − sin theta). In terms of the sagitta h, A = R^2 arccos((R − h)/R) − (R − h) sqrt(2Rh − h^2). The latter formula follows from the circle sector minus the isosceles triangle formed by the radii to the segment’s endpoints.

Notes: A Kreissegment is the difference between the area of the corresponding circle sector and the area of the chord’s triangle; it is distinct from the sector itself, which is bounded by radii and the arc. Kreissegmente have uses in geometry, design, and shading problems where curved boundaries are involved.