Central Angle
GeometryA 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 is an angle formed at the center of a circle. Its two sides are radii (lines from the center to the edge). The central angle has the same number of degrees as the arc it cuts off.
Example
If a central angle is $90^\circ$, it cuts off a $90^\circ$ arc (one quarter of the circle). A central angle of $180^\circ$ cuts off a semicircle. A slice of pie cut from the center forms a central angle.
Key Insight
The central angle and its arc always have the same degree measure - that is the definition of arc measure. A full circle is $360^\circ$, so a central angle of $1^\circ$ cuts off $1/360$ of the circle.
Definition
A central angle has its vertex at the center of the circle and its sides along two radii. The measure of a central angle equals the degree measure of the intercepted arc. A sector (pie slice) is the region bounded by two radii and the intercepted arc. The area of a sector $= (\theta/360)\pi r^2$.
Example
Central angle of $72^\circ$: arc measure $= 72^\circ$, arc length $= (72/360) \times 2\pi r = (1/5) \times 2\pi r = 2\pi r/5$. Sector area $= (72/360)\pi r^2 = \pi r^2/5$. For $r=10$: arc length $= 4\pi$ approximately $12.57$, sector area $= 20\pi$ approximately $62.83$.
Key Insight
The inscribed angle theorem compares central and inscribed angles: for the same arc, the inscribed angle is exactly half the central angle. So if a central angle is $80^\circ$, any inscribed angle intercepting the same arc is $40^\circ$.
Definition
A central angle in radians is the ratio of arc length to radius: $\theta = s/r$. This is the definition of radian measure. The central angle subtended by a chord of length $c$ in a circle of radius $r$ is $\theta = 2\arcsin(c/(2r))$. In complex analysis, the central angle argument relates to the argument (phase) of a complex number on the unit circle.
Example
Arc length $s = 8$, radius $r = 5$: central angle $\theta = 8/5 = 1.6$ radians $= 91.67^\circ$. Chord for this central angle: $c = 2\cdot5\sin(0.8) = 10\sin(0.8)$ approximately $7.17$.
Key Insight
The radian definition of angle ($\theta = \text{arc}/\text{radius}$) makes the central angle dimensionless and ties angle directly to arc length. This dimensionlessness is why calculus formulas for trigonometric functions ($d/dx\sin x = \cos x$) require radians: the derivative uses the small-angle approximation $\sin x \approx x$, valid only when $x$ is in radians.