Arc length is defined as the length along a curve,

(1) 
Defining the line element
, parameterizing the curve in terms of a parameter , and noting
that is simply the magnitude of the Velocity with which the end of the Radius Vector
moves gives

(2) 
In Polar Coordinates,

(3) 
so
In Cartesian Coordinates,
Therefore, if the curve is written

(8) 
then

(9) 
If the curve is instead written

(10) 
then

(11) 
Or, in three dimensions,

(12) 
so

(13) 
See also Curvature, Geodesic, Normal Vector, Radius of Curvature, Radius of Torsion,
Speed, Surface Area, Tangential Angle, Tangent Vector, Torsion (Differential Geometry),
Velocity
© 19969 Eric W. Weisstein
19990525