Arc Length (Radian)
TrigonometryArc 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 if you walk along the curved edge of a circle. The formula is simple: arc length $=$ radius $\times$ angle (in radians).
Example
A circle has radius $6$. The central angle is $\pi/3$ radians ($60^\circ$). Arc length $= 6 \times \pi/3 = 2\pi \approx 6.28$ units.
Key Insight
This formula is why radians exist. With degrees, the formula is messy. With radians: $s = r\theta$, as clean as it gets.
Definition
For a circle of radius $r$ with a central angle of $\theta$ radians, the arc length is $s = r\theta$. This comes from the proportion: $s/\text{circumference} = \theta/(2\pi)$, so $s = r\theta$. If $\theta$ is in degrees, multiply by $\pi/180$ first.
Example
A pizza with radius $15$ cm is cut at an angle of $40^\circ = 40 \times (\pi/180) = 2\pi/9$ radians. The crust length $= 15 \times (2\pi/9) \approx 15 \times 0.698 \approx 10.47$ cm.
Key Insight
On the unit circle ($r = 1$), arc length equals the angle in radians. This is the deepest reason why radian measure is natural: the angle IS the arc length on the unit circle.
Definition
The arc length formula $s = r\theta$ is the simplest case of the general arc length integral: $L = \int_a^b \sqrt{1 + (dy/dx)^2}\, dx$. For a parametric circle $x = r\cos(t)$, $y = r\sin(t)$, this reduces to $L = r(b - a) = r\theta$, recovering the formula. The radian definition of arc length is foundational to the theory of curves in differential geometry.
Example
In a unit-circle parametrization, arc length from angle $0$ to angle $\theta$ is exactly $\theta$. This means radian measure parametrizes the circle by arc length, making it the natural "arc length parameter" used in differential geometry.
Key Insight
A curve parametrized by arc length (unit-speed parametrization) is the most natural choice in differential geometry. 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.