# Circular sector

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, the radius of the circle, and 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 . The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle and (because{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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}} the area of the sector is proportional to the angle, and is the angle for the whole circle, in radians):

The area of a sector in terms of can be obtained by multiplying the total area by the ratio of to the total perimeter .

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

Converting the central angle into degrees gives

## Perimeter

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

where *θ* is in radians.

## See also

- Circular segment - the part of the sector which remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
- Conic section

## References

- Gerard, L. J. V.
*The Elements of Geometry, in Eight Books; or, First Step in Applied Logic*, London, Longman's Green, Reader & Dyer, 1874. p. 285

## External links

- Definition and properties of a circle sector with interactive animation
- Weisstein, Eric W., "Circular sector",
*MathWorld*.