# 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

$\{\begin{array}{cc}a=r\alpha ,\hfill & \\ c=\mathrm{\hspace{0.33em}2}r\mathrm{sin}\frac{\alpha}{2}.\hfill & \end{array}$ | (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 $\frac{c}{a}\cdot \frac{\alpha}{2}=\mathrm{sin}\frac{\alpha}{2}$ 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^{}]:

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

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:

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

Title | circular segment |
---|---|

Canonical name | CircularSegment |

Date of creation | 2013-03-22 19:05:02 |

Last modified on | 2013-03-22 19:05:02 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 26B10 |

Classification | msc 51M04 |

Related topic | LineSegment |

Related topic | SphericalSegment |

Related topic | ExampleOfCalculusOfVariations |

Defines | height of circular segment |