The arc length of the curve is given by .

Let s(t) represent the arc length which the particle has moved along the curve.

The curve is thus parametrized in a preferred manner by its arc length.

The formula for arc length is a bit messy, but unambiguous.

You can find terms like perimeter, arc length, area discussed at all levels of sophistication on a "Google" search.

The perimeter (total arc length) of the deltoid is 16a.

This implies that no curve can have an arc length less than the distance between its endpoints.

The arc length of a curve on the surface and the surface area can be found using integration.

The circle is the plane curve enclosing the maximum area for a given arc length.

In terms of arc length s let the path be described as: