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Arc Length of a Curve

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:

Arc length equals the integral from a to b of ds
 

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:

Formula: ds equals the square root of 1 plus (dy/dx) squared, multiplied by dx

ds = square root of (1 + (dx/dy)²) dy

2. In parametric form, use

Formula: ds = sqrt((dx/dt)² + (dy/dt)²) dt, used to find arc length in parametric form.

3. In polar form, use

Formula: ds equals the square root of r squared plus (dr/dθ) squared, times dθ

 
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.

Graph of y = (1/6)x³ + (1/2)x⁻¹ with a highlighted red arc segment between x = 1 and x = 2.

Step-by-step arc length calculation: integral from 1 to 2 of sqrt(1+(dy/dx)²)dx, simplifying to 17/12.

 
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.

A red arc (semicircle) with maximum height 2, plotted from t=0 to t=2π, with x-axis labels at 2, 4, 6.

Arc length of a cycloid

 
Example 3: Polar Find the length of the first rotation of the logarithmic spiral r = eθ. The values of θ run from 0 to 2π.

A curve plotted on a coordinate grid spanning x: -100 to 500, y: -100 to 100, showing a parabolic arc opening rightward.

Arc length formula in polar form: integral from 0 to 2π of √((e^θ)²+(e^θ)²)dθ, simplified to √2(e^2π−1)

 

See also

Surface area of a surface of revolution