Central Angle
A central angle is an angle whose vertex is at the center of a circle, with sides that are radii, and whose measure equals the arc it intercepts.
Formula
\text{central angle} = \text{arc measure (in degrees)}
Definition
A central angle has its vertex at the center of a circle with its sides along two radii; its measure always equals the degree measure of the arc it intercepts. In radians, a central angle is defined as the ratio of arc length to radius, $\theta=s/r$, which is also the definition of radian measure; the central angle subtended by a chord of length $c$ is $\theta=2\arcsin(c/(2r))$. A sector (pie slice) is bounded by two radii and the intercepted arc, with area $(\theta/360)\pi r^2$ in degrees.
Example
A central angle of $90^\circ$ cuts off a quarter of the circle, and $180^\circ$ cuts off a semicircle; a slice of pie cut from the center forms a central angle. For a central angle of $72^\circ$: arc measure $=72^\circ$, arc length $=2\pi r/5$, sector area $=\pi r^2/5$; for $r=10$, arc length $\approx12.57$ and sector area $\approx62.83$. For arc length $8$ and radius $5$: central angle $=8/5=1.6$ radians $\approx91.67^\circ$, corresponding to a chord of $10\sin(0.8)\approx7.17$.
Key Insight
A central angle and its arc always share the same degree measure, that is the very definition of arc measure, and by the inscribed angle theorem, an inscribed angle intercepting the same arc is always exactly half the central angle. The radian definition, $\theta=$ arc/radius, makes the angle dimensionless and ties it directly to arc length, which is why calculus formulas for trig derivatives require radians, since they rely on the small-angle approximation $\sin x\approx x$, only valid when $x$ is measured in radians.