Segment

A portion of a Circle whose upper boundary is a circular Arc and whose lower boundary is a Chord making a Central Angle $\theta<\pi$ radians (180°), illustrated above as the shaded region. Let $R$ be the radius of the Circle, $c$ the Chord length, $s$ the Arc Length, $h$ the height of the arced portion, and $d$ the height of the triangular portion. Then

$$R = h+d$$ (1)

$$s = R\theta$$ (2)

$$d = R\cos({\textstyle{1\over 2}}\theta)$$ (3)

$$= {\textstyle{1\over 2}}c\cot({\textstyle{1\over 2}}\theta)$$ (4)

$$= {\textstyle{1\over 2}}\sqrt{4R^2-c^2}$$ (5)

$$c = 2R\sin({\textstyle{1\over 2}}\theta)$$ (6)

$$= 2d\tan({\textstyle{1\over 2}}\theta)$$ (7)

$$= 2\sqrt{R^2-d^2}$$ (8)

$$= 2\sqrt{h(2R-h)}\,.$$ (9)

The Angle $\theta$ obeys the relationships

$$\theta = {s\over R}=2\cos^{-1}\left({d\over R}\right)=2\tan^{-1}\left({c\over 2d}\right)$$

$$= 2\sin^{-1}\left({c\over 2R}\right).$$ (10)

The Area of the segment is then

$$A = A_{\rm sector}-A_{\rm isosceles\ triangle}$$ (11)

$$= {\textstyle{1\over 2}}R^2(\theta-\sin\theta)$$ (12)

$$= {\textstyle{1\over 2}}(Rs-cd)$$ (13)

$$= R^2\cos^{-1}\left({d\over R}\right)-d\sqrt{R^2-d^2}$$ (14)

$$= R^2\cos^{-1}\left({R-h\over R}\right)-(R-h)\sqrt{2Rh-h^2},$$ (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.