Arc
An arc is a portion of the circumference of a circle, defined by two endpoints and measured in degrees or in length.
Formula
\text{Arc length} = (\theta/360) \times 2\pi r \text{ (}\theta \text{ in degrees)}
Definition
An arc is a curved part of a circle between two points on it; a minor arc is the smaller piece, a major arc the larger piece, and a semicircle is exactly half. Arc length $=(\theta/360)\times2\pi r$ for central angle $\theta$ in degrees, or simply $r\theta$ when $\theta$ is in radians, since an arc is a rectifiable curve whose length is the integral of $|r'(t)|\,dt$ over its parameter interval; for a circle parametrized as $(r\cos t, r\sin t)$, arc length from $t=0$ to $t=\theta$ works out to exactly $r\theta$.
Example
Marking two points on a circle creates two arcs, one short and one long, like a rainbow forming an arc; a quarter-circle arc spans exactly $90^\circ$. For radius $6$ and central angle $120^\circ$: arc length $=(120/360)\times2\pi\cdot6=4\pi\approx12.57$, with the arc's degree measure equal to the central angle, $120^\circ$. For an ellipse, arc length has no closed form and requires an elliptic integral, unlike the simple $r\theta$ formula for a circle.
Key Insight
An arc can be measured either as an angle (degrees from the center) or as a length (actual distance along the curve), and radians make the length formula especially clean, arc length $=r\theta$ with no conversion factor, which is why radians are the "natural" unit for circular measurement. The failure of this simple formula for other conics motivated the development of elliptic functions by Abel and Jacobi, giving rise to elliptic curves, objects fundamental to modern number theory and cryptography, and central to Wiles' proof of Fermat's Last Theorem.