# 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

 $\displaystyle\begin{cases}a\;=\;r\alpha,\\ c\;=\;2r\sin\frac{\alpha}{2}.\end{cases}$ (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:

 $h\;=\;\left(1-\cos\frac{\alpha}{2}\right)r\;=\;r-\sqrt{r^{2}-\frac{c^{2}}{4}}% \;=\;\frac{c}{2}\tan\frac{\alpha}{4}$
Title circular segment CircularSegment 2013-03-22 19:05:02 2013-03-22 19:05:02 pahio (2872) pahio (2872) 10 pahio (2872) Definition msc 26B10 msc 51M04 LineSegment SphericalSegment ExampleOfCalculusOfVariations height of circular segment