Circular segment

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In geometry, a circular segment (symbol: ) is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc (of less than 180°) of a circle and by the chord connecting the endpoints of the arc.


A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area).

Let R be the radius of the circle, θ is the central angle in radians, α is the central angle in degrees, c the chord length, s the arc length, h the sagitta (height) of the segment, and d the height of the triangular portion.

  • The radius is R = h + d = h/2+c^2/8h \frac{}{}
  • The arc length is s = \frac{\alpha}{180}\pi R = {\theta} R = \arcsin\left(\frac{c}{h+\frac{c^2}{4h}}\right)\left(h + \frac{c^2}{4h}\right)
  • The chord length is c = 2R\sin\frac{\theta}{2} = R\sqrt{2-2\cos\theta}
  • The sagitta is h = R\left(1-\cos\frac{\theta}{2}\right) = R - \sqrt{R^2 - \frac{c^2}{4}}
  • The angle is  \theta = 2\arctan\frac{c}{2d}= 2\arccos\frac{d}{R} = 2\arcsin\frac{c}{2R}


The area of the circular segment is equal to the area of the circular sector minus the area of the triangular portion—that is,

A =  \frac{R^2}{2} \left(\theta - \sin \theta \right).

or with the central in degrees,

A =  \frac{R^2}{2} \left(\frac{\alpha\pi}{180} - \sin \alpha \right)

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