arc length of a portion of a circle is angular distance.
arc length of a portion of a circle?
angular distance
let "r" be the radius of the circle, "θ" the central angle in radians, α is the central angle in degrees, "c" the chord length, "s" the arc length, "h" the sagitta (height) of the segment, and "d" the height of the triangular portion.





























the radius is





























the radius in terms of "h" and "c" can be derived above by using the intersecting chords theorem, where 2"r" (the diameter) and "c" are perpendicularly intersecting chords:





























the arc length is
circular segment
note that when formula_13, that "u" then equals the quarter period "k".





























section::::definition as trigonometry - the jacobi ellipse.





























formula_14 are defined on the unit circle, with radius "r" = 1 and angle formula_15 arc length of the unit circle measured from the positive "x"-axis. similarly, jacobi elliptic functions are defined on the unit ellipse, with "a" = 1. let





























then:





























for each angle formula_11 the angular component of the arc length, the





























'angular arc length' is computed. an advantage of angular arc length is that we can calculate a total arc length





























for hyperbolas. for the ellipse the angular arc length is:
jacobi elliptic functions