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# circular segment

A chord of a circle divides 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 $c$ ($<a$) uniquely determine $r$ and $\alpha$ from the pair of equations (1), but $r$ and $\alpha$ are generally not expressible 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 [the adding concerns the case where the central angle is greater than the straight angle]:

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

## Mathematics Subject Classification

26B10*no label found*51M04

*no label found*

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