Arc
GeometryAn 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 the circle. A minor arc is the smaller curved piece; a major arc is the larger curved piece. A semicircle is a half arc.
Example
If you mark two points on a circle, you get two arcs - one short and one long. Like a rainbow is an arc of a circle. A quarter-circle arc spans exactly $90^\circ$.
Key Insight
An arc is measured two ways: as an angle (how many degrees it spans from the center) or as a length (the actual distance along the curve). The degree measure is simpler; the length depends on the circle's size too.
Definition
An arc is a connected portion of a circle's circumference. A minor arc spans less than $180^\circ$; a major arc spans more than $180^\circ$; a semicircle spans exactly $180^\circ$. Arc length $= (\theta/360) \times 2\pi r$, where $\theta$ is the central angle in degrees. Arc degree measure = central angle measure.
Example
Circle with radius $6$, central angle $120^\circ$: arc length $= (120/360) \times 2\pi\cdot6 = (1/3) \times 12\pi = 4\pi$ approximately $12.57$. The arc degree measure is $120^\circ$ (same as the central angle).
Key Insight
Arc length formula in radians is simpler: arc length $= r\theta$ (where $\theta$ is in radians). This is why radians are the "natural" unit for circular measurement - the arc length is simply radius times angle, with no conversion factor needed.
Definition
An arc of a circle is a rectifiable curve: its length is the integral of $|r'(t)|\,dt$ over the parameter interval. For a circle of radius $r$ parametrized as $r(t) = (r\cos t, r\sin t)$, arc length from $t=0$ to $t=\theta$ is $\int_0^\theta r\,dt = r\theta$. The arc length formula generalizes to the integral definition of arc length for smooth curves in $\mathbb{R}^n$.
Example
For an ellipse with semi-axes $a$ and $b$, arc length requires an elliptic integral (no closed form). This contrasts with the circle, where arc length $= r\theta$. The incomputability of elliptic arc lengths in elementary functions motivated the development of elliptic functions by Abel and Jacobi.
Key Insight
The simple arc length formula for circles (length $= r\theta$) fails for all other conics. The resulting elliptic integrals, though "non-elementary," are deeply structured and gave rise to elliptic curves - fundamental objects in number theory used in Wiles' proof of Fermat's Last Theorem and in modern cryptography.