Arc Length of Curve in Polar Coordinates

Theorem

Let $C$ be a curve embedded in a polar plane.

As a Function of Radial Coordinate

Let the angular coordinate $\theta$ of $C$ be defined as a function of the radial coordinate $r$:

$\theta = \map f r$

The arc length $s$ of $C$ between $r = u$ and $r = v$ is defined as:

$\ds s := \int_u^v \paren {\sqrt {1 + r^2 \paren {\frac {\d \theta} {\d r} }^2} } \rd r$

As a Function of Angular Coordinate

Let the radial coordinate $r$ of $C$ be defined as a function of the angular coordinate $\theta$:

$r = \map f \theta$

The arc length $s$ of $C$ between $\theta = \alpha$ and $\theta = \beta$ is defined as:

$\ds s := \int_\alpha^\beta \paren {\sqrt {\paren {\frac {\d r} {\d \theta} }^2 + r^2} } \rd \theta$