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:

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

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