User:Eirik1231/Circular segment

In geometry, a circular segment (symbol: ⌓) is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord. The circle segment constitutes the part between the secant and an arc, excluding of the circle's center.

Formulas


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 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 $$s = \frac{\alpha}{360}\tau R = {\theta} R$$


 * The chord length is $$c = 2R\sin\frac{\theta}{2} = R\sqrt{2-2\cos\theta}$$


 * The height is $$h = R(1-\cos\frac{\theta}{2}) = R - \sqrt{R^2 - \frac{c^2}{4}} $$


 * The angle is $$ \theta = 2\arccos\frac{d}{R} = 2\arcsin\frac{c}{2R}$$

Area
The area of the circular segment is equal to the area of the circular sector minus the area of the triangular portion.

$$A = {1 \over 2} \tau R^2 \cdot \frac{\theta}{\tau} - \frac {R^2 \sin \theta}{2} = \frac{R^2}{2} \left(\theta - \sin\theta \right) = \frac{R^2}{2} \left( \frac {\alpha\tau}{360} - \sin \frac{\alpha\tau}{360}\right)$$