Arc length is defined as the length along a curve,

where is a differential displacement vector along a curve . For example, for a circle of radius , the arc length between two points with angles and (measured in radians) is simply

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

In polar coordinates,

so

In Cartesian coordinates,

Therefore, if the curve is written

then

If the curve is instead written

then

In three dimensions,

so

The arc length of the polar curve is given by

See also

Curvature, Geodesic, Normal Vector, Radius of Curvature, Radius of Torsion, Speed, Surface Area, Tangent Vector, Tangential Angle, Torsion, Velocity Explore this topic in the MathWorld classroom

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Arc Length." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ArcLength.html

Subject classifications