circular segment

A chord of a circle the corresponding disk into two circular segments.  The perimetre of a circular segment consists thus of the chord ($c$) and a circular arc ($a$).

The magnitude $r$ of the radius of circle and the magnitude $\alpha$ of a central angle naturally determine uniquely the magnitudes of the corresponding arc and chord, and these may be directly calculated from

$\{\begin{array}{cc}a=r\alpha ,\hfill & \\ c=\mathrm{\hspace{0.33em}2}r\sin \frac{\alpha }{2}.\hfill & \end{array}$(1)

Conversely, the magnitudes of $a$ and $c$ () uniquely determine $r$ and $\alpha$ from the pair of equations (1), but $r$ and $\alpha$ are generally not in a closed form; this becomes clear from the relationship  $\frac{c}{a}\cdot \frac{\alpha }{2}=\sin \frac{\alpha }{2}$  implied by (1).

The area of a circular segment is obtained by subtracting from [resp. adding to] the area of the corresponding sector the area of the isosceles triangle having the chord as base (http://planetmath.org/BaseAndHeightOfTriangle) [the adding concerns the case where the central angle is greater than the straight angle]:

$$A=\frac{\alpha }{2\pi }\cdot \pi r^2\mp \frac{1}{2}{r}^2\sin \alpha =\frac{r^2}{2}(\alpha \mp \sin \alpha )$$

The of the circular segment, i.e. the distance of the midpoints (http://planetmath.org/ArcLength) of the arc and the chord, may be expressed in the following forms:

$$h=\left(1-\cos \frac{\alpha }{2}\right)r=r-\sqrt{r^2-\frac{c^2}{4}}=\frac{c}{2}\tan \frac{\alpha }{4}$$