## Segment

A portion of a Circle whose upper boundary is a circular Arc and whose lower boundary is a Chord making a Central Angle radians (180°), illustrated above as the shaded region. Let be the radius of the Circle, the Chord length, the Arc Length, the height of the arced portion, and the height of the triangular portion. Then

 (1) (2) (3) (4) (5) (6) (7) (8) (9)

The Angle obeys the relationships
 (10)

The Area of the segment is then
 (11) (12) (13) (14) (15)

where the formula for the Isosceles Triangle in terms of the Vertex angle has been used (Beyer 1987).

See also Chord, Circle-Circle Intersection, Cylindrical Segment, Lens, Parabolic Segment, Sagitta, Sector, Spherical Segment

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 125, 1987.