Arc length

Arc length is the distance between two points along a section of a curve.

Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length).

If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) ${\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}}$, then the curve is rectifiable (i.e., it has a finite length).

The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.