Arc Length
Arc length is the distance along a curved portion of a circle, proportional to the central angle and the radius.
Formula
s = r\theta \text{ (radians) or } s = \frac{\theta}{360} \cdot 2\pi r \text{ (degrees)}
Definition
An arc is a curved piece of a circle, like a slice of the rim, and arc length is how long that curved piece is: for a circle of radius $r$ and a central angle of $\theta$ degrees, the arc length is $s = (\theta/360) \cdot 2\pi r$, which in radians simplifies to $s = r\theta$. Arc length is always proportional to the central angle, so a half-circle arc is always half the circumference and a quarter-circle arc is always a quarter. More generally, for a smooth curve parameterized by $r(t) = (x(t), y(t))$ on $[a, b]$, the arc length is $$L = \int_a^b |r'(t)| \, dt = \int_a^b \sqrt{x'(t)^2 + y'(t)^2} \, dt,$$ which for a circle of radius $R$ parameterized as $(R\cos t, R\sin t)$ on $[0, \theta]$ gives $L = R\theta$.
Example
If a circle has circumference $60$ cm and you take a quarter of it ($90$ out of $360$ degrees), the arc length is $(90/360) \times 60 = 15$ cm. A circle of radius $9$ cm with a central angle of $120$ degrees gives $s = (120/360) \cdot 2\pi \cdot 9 = (1/3)(18\pi) = 6\pi = 18.85$ cm; in radians, $120$ degrees $= 2\pi/3$, so $s = 9 \cdot (2\pi/3) = 6\pi$, matching exactly. The arc length of $y = x^{3/2}$ from $x=0$ to $x=4$ is $L = \int_0^4 \sqrt{1 + (3/2\sqrt{x})^2} \, dx = \int \sqrt{1 + 9x/4} \, dx$, evaluating to $(8/27)(10\sqrt{10} - 1) \approx 9.07$.
Key Insight
The radian formula $s = r\theta$ is simpler and more natural than the degree formula, since it becomes a direct product without conversion factors, one reason radians are preferred in higher mathematics. The arc length integral is not always solvable in closed form; for ellipses it leads to elliptic integrals, which cannot be expressed with elementary functions, motivating entire new branches of analysis in the 18th and 19th centuries.