Sector
GeometryA 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" part of a circle, bounded by two radii and an arc. If you cut a pizza into slices, each slice is a sector of the circular pizza.
Example
A quarter-circle is a sector with a $90^\circ$ central angle. If the whole pizza (radius $8$ inches) has area $64\pi$ approximately $201$ square inches, one-quarter of it (a sector with $90^\circ$ angle) has area about $50.3$ square inches.
Key Insight
A sector is to a circle what a slice is to a pie. The fraction of the full circle you get depends on the central angle: a $90^\circ$ sector is $90/360 =$ one quarter of the circle.
Definition
A sector is the region bounded by two radii and the intercepted arc. For central angle $\theta$ (in degrees): area $= (\theta/360)\pi r^2$. In radians: area $= (1/2)r^2\theta$. Arc length $= r\theta$ (radians). A sector is a minor sector (central angle $< 180$) or major sector (central angle $> 180$).
Example
Sector with radius $6$ and central angle $120^\circ$: area $= (120/360) \times 36\pi = (1/3) \times 36\pi = 12\pi$ approximately $37.7$. Arc length $= (120/360) \times 2\pi\cdot6 = 4\pi$ approximately $12.57$. In radians: $\theta = 2\pi/3$, area $= (1/2) \times 36 \times (2\pi/3) = 12\pi$. Same answer.
Key Insight
The area formula $(1/2)r^2\theta$ (in radians) has a beautiful analogy: $(1/2) \times \text{base} \times \text{height}$ for a triangle, where base $=$ arc $= r\theta$ and height $= r$. A sector is like an "unrolled" triangle with a curved base - and the same formula structure applies.
Definition
A sector with central angle $\theta$ (radians) and radius $r$ has area $A = (1/2)r^2\theta$, which can be derived by integrating: $A = \int_0^\theta (1/2)r^2\,d\phi = (1/2)r^2\theta$. The sector generalizes to the "wedge" in polar coordinates: the area between two curves $r=f(\phi)$ and $r=g(\phi)$ is $\int_\alpha^\beta (1/2)(f^2 - g^2)\,d\phi$.
Example
Polar area for $r = 2\cos\theta$ from $0$ to $\pi/2$: $A = \int_0^{\pi/2} (1/2)(2\cos\theta)^2\,d\theta = 2\int_0^{\pi/2} \cos^2\theta\,d\theta = 2(\pi/4) = \pi/2$. This traces the sector of the circle $x^2+y^2=2x$ (circle of radius $1$, centered at $(1,0)$).
Key Insight
The polar area formula $A = (1/2)\int r^2\,d\theta$ is the natural generalization of sector area to arbitrary polar curves. It is one of the most useful integration techniques in calculus, appearing in problems about rose curves, limacons, and other polar figures. The factor of $1/2$ is the same "$1/2$" as in the triangle area and sector area formulas, reflecting their shared infinitesimal geometry.