# Circular sector

A circular sector is shaded in green

A circular sector or circle sector (symbol: ), is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, ${\displaystyle r}$ the radius of the circle, and ${\displaystyle L}$ is the arc length of the minor sector.

A sector with the central angle of 180° is called a semicircle. Sectors with other central angles are sometimes given special names, these include quadrants (90°), sextants (60°) and octants (45°).

The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle.

## Area

The total area of a circle is ${\displaystyle \pi r^{2}}$. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle and ${\displaystyle 2\pi }$ (because{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} the area of the sector is proportional to the angle, and ${\displaystyle 2\pi }$ is the angle for the whole circle, in radians):

${\displaystyle A=\pi r^{2}\cdot {\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}}$

The area of a sector in terms of ${\displaystyle L}$ can be obtained by multiplying the total area ${\displaystyle \pi r^{2}}$by the ratio of ${\displaystyle L}$ to the total perimeter ${\displaystyle 2\pi r}$.

${\displaystyle A=\pi r^{2}\cdot {\frac {L}{2\pi r}}={\frac {r\cdot L}{2}}}$

Another approach is to consider this area as the result of the following integral :

${\displaystyle A=\int _{0}^{\theta }\int _{0}^{r}dS=\int _{0}^{\theta }\int _{0}^{r}{\tilde {r}}d{\tilde {r}}d{\tilde {\theta }}=\int _{0}^{\theta }{\frac {1}{2}}r^{2}d{\tilde {\theta }}={\frac {r^{2}\theta }{2}}}$

Converting the central angle into degrees gives

${\displaystyle A=\pi r^{2}\cdot {\frac {\theta ^{\circ }}{360}}}$

## Perimeter

The length of the perimeter of a sector is the sum of the arc length and the two radii:

${\displaystyle P=L+2r=\theta r+2r=r\left(\theta +2\right)}$