circular segment

A chord of a circle the corresponding disk into two circular segmentsMathworldPlanetmath.  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 α of a central angleMathworldPlanetmath naturally determine uniquely the magnitudes of the corresponding arc and chord, and these may be directly calculated from

{a=rα,c= 2rsinα2. (1)

Conversely, the magnitudes of a and c (<a) uniquely determine r and α from the pair of equations (1), but r and α are generally not in a closed form; this becomes clear from the relationship  caα2=sinα2  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 triangleMathworldPlanetmath having the chord as base ( [the adding concerns the case where the central angle is greater than the straight angleMathworldPlanetmath]:


The of the circular segment, i.e. the distanceMathworldPlanetmath of the midpointsMathworldPlanetmathPlanetmathPlanetmath ( of the arc and the chord, may be expressed in the following forms:

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