Arc Length (Radian)
Arc length in radian measure is the distance along a circular arc, calculated as the product of the radius and the central angle in radians.
Formula
s = r\theta \text{ (where } \theta \text{ is in radians)}
Definition
Arc length is the distance you travel walking along the curved edge of a circle, given by the simple formula $s = r\theta$, radius times angle in radians. For a circle of radius $r$ with a central angle of $\theta$ radians, this comes from the proportion $s/\text{circumference} = \theta/(2\pi)$; if $\theta$ is given in degrees, multiply by $\pi/180$ first. This is the simplest case of the general arc length integral $L = \int_a^b \sqrt{1 + (dy/dx)^2}\, dx$, which for a parametric circle $x = r\cos(t)$, $y = r\sin(t)$ reduces exactly to $L = r\theta$.
Example
A circle has radius $6$ and central angle $\pi/3$ radians ($60^\circ$); the arc length $= 6 \times \pi/3 = 2\pi \approx 6.28$ units. A pizza with radius $15$ cm cut at $40^\circ = 2\pi/9$ radians has crust length $= 15 \times (2\pi/9) \approx 10.47$ cm. In a unit-circle parametrization, arc length from angle $0$ to angle $\theta$ is exactly $\theta$, showing that radian measure parametrizes the circle by arc length, the natural "arc length parameter" used in differential geometry.
Key Insight
This formula is why radians exist: with degrees the formula is messy, but with radians, $s = r\theta$ is as clean as it gets, and on the unit circle ($r = 1$), arc length equals the angle in radians directly. A curve parametrized by arc length (unit-speed parametrization) is the most natural choice in differential geometry, and the unit circle parametrized by $t \in [0, 2\pi)$ is automatically arc-length parametrized, connecting the radian angle directly to the geometric speed concept.