Circular segment

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.

Formula

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 $\theta = 2\arctan\frac{c}{2d}= 2\arccos\frac{d}{R} = 2\arccos\left(1- \frac{h}{R}\right) = 2\arcsin\frac{c}{2R}$

Area

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

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

with the central angle in radians, or

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

with the central angle in degrees.

See also

• Spherical segment
• Conic section
• Cross section

References

• Weisstein, Eric W., "Circular segment", MathWorld.

External links

• Definition of a circular segment With interactive animation
• Formulae for area of a circular segment With interactive animation