# 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, $r$ the radius of the circle, and $L$ 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 $\pi r^2$. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle and $2 \pi$ (because the area of the sector is proportional to the angle, and $2 \pi$ is the angle for the whole circle):

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

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

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

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

$A = \int_0^\theta\int_0^r dS=\int_0^\theta\int_0^r \tilde{r} d\tilde{r} d\tilde{\theta} = \int_0^\theta \frac{1}{2} r^2 d\tilde{\theta} = \frac{r^2 \theta}{2}$

Converting the central angle into degrees gives

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

## Perimeter

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

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