Mathwords: Arc Length of a Curve

index: click on a letter
A	B	C	D	E
F	G	H	I	J
K	L	M	N	O
P	Q	R	S	T
U	V	W	X	Y
Z	A to Z index
index: subject areas
numbers & symbols
sets, logic, proofs
geometry
algebra
trigonometry
advanced algebra & pre-calculus
calculus
advanced topics
probability & statistics
real world applications

multimedia entries

www.mathwords.com	about mathwords
	website feedback

Arc Length of a Curve

The length of a curve or line.

The length of an arc can be found by one of the formulas below for any differentiable curve defined by rectangular, polar, or parametric equations.

For the length of a circular arc, see arc of a circle.

Formula:
	where a and b represent x, y, t, or θ-values as appropriate, and ds can be found as follows. 1. In rectangular form, use whichever of the following is easier: 2. In parametric form, use 3. In polar form, use
Example 1: Rectangular	Find the length of an arc of the curve y = (1/6) x³ + (1/2) x^–1 from
	x = 1 to x = 2.
Example 2: Parametric	Find the length of the arc in one period of the cycloid x = t – sin t, y = 1 – cos t. The values of t run from 0 to 2π.
Example 3: Polar	Find the length of the first rotation of the logarithmic spiral r = e^θ. The values of θ run from 0 to 2π.

See also

Surface area of a surface of revolution