Sector
A sector is the region of a circle bounded by two radii and the arc between them, shaped like a pie slice.
Formula
\text{Area} = (\theta/360)\pi r^2 = (1/2)r^2\theta \text{ (}\theta \text{ in radians)}
Definition
A sector is the "pie slice" region of a circle bounded by two radii and the arc between them. For central angle $\theta$ in degrees, area $=(\theta/360)\pi r^2$; in radians, area $=(1/2)r^2\theta$ and arc length $=r\theta$, and a sector is called minor (central angle less than $180^\circ$) or major (greater than $180^\circ$). The radian-based area formula can be derived by integration, $A=\int_0^\theta(1/2)r^2\,d\phi=(1/2)r^2\theta$, and it generalizes to the polar-coordinate area formula $A=(1/2)\int r^2\,d\theta$ used for any polar curve.
Example
A quarter-circle is a sector with a $90^\circ$ central angle; a pizza of radius $8$ inches (area $64\pi\approx201$ square inches) has a $90^\circ$ slice of about $50.3$ square inches. For radius $6$ and central angle $120^\circ$: area $=(120/360)\times36\pi=12\pi\approx37.7$, arc length $=4\pi\approx12.57$; in radians ($\theta=2\pi/3$), area $=(1/2)(36)(2\pi/3)=12\pi$, the same answer. For the polar curve $r=2\cos\theta$ from $0$ to $\pi/2$: area $=\int_0^{\pi/2}(1/2)(2\cos\theta)^2\,d\theta=2(\pi/4)=\pi/2$, tracing a sector of the circle $x^2+y^2=2x$.
Key Insight
A sector is to a circle what a slice is to a pie, and the fraction of the full circle depends on the central angle: a $90^\circ$ sector is one quarter of the circle. The radian area formula $(1/2)r^2\theta$ has a beautiful analogy to $(1/2)\times$base$\times$height for a triangle, with base $=$ arc $=r\theta$ and height $=r$, as though the sector were an "unrolled" triangle with a curved base, and this same $1/2$ resurfaces in the polar area formula, one of the most useful integration techniques in calculus for rose curves, limacons, and other polar figures.