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.^{[1]}^{:234} In the diagram, θ is the central angle, 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 half-disk and is bounded by a diameter and a semicircle. Sectors with other central angles are sometimes given special names, these include **quadrants** (90°), **sextants** (60°) and **octants** (45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively. Confusingly, the arc of a quadrant can also be termed a quadrant.

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.^{[2]}^{:376}

## Area

The total area of a circle is π*r*^{2}. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and 2π (because the area of the sector is directly proportional to its angle, and 2π is the angle for the whole circle, in radians):

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

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

Converting the central angle into degrees gives^{[3]}

## 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.

## Arc length

The formula for the length of an arc is:^{[4]}^{:570}

where L represents the arc length, r represents the radius of the circle and θ represents the angle in radians made by the arc at the centre of the circle.^{[5]}^{:79}

If the value of angle is given in degrees, then we can also use the following formula by:^{[3]}

## Chord length

The length of a chord formed with the extremal points of the arc is given by

where C represents the chord length, R represents the radius of the circle, and θ represents the angular width of the sector 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

**^**Dewan, R. K.,*Saraswati Mathematics*(New Delhi: New Saraswati House, 2016), p. 234.**^**Achatz, T., & Anderson, J. G., with McKenzie, K., ed.,*Technical Shop Mathematics*(New York: Industrial Press, 2005), p. 376.- ^
^{a}^{b}Uppal, Shveta (2019).*Mathematics: Textbook for class X*. New Delhi: NCERT. pp. 226, 227. ISBN 81-7450-634-9. OCLC 1145113954. **^**Larson, R., & Edwards, B. H.,*Calculus I with Precalculus*(Boston: Brooks/Cole, 2002), p. 570.**^**Wicks, A.,*Mathematics Standard Level for the International Baccalaureate*(West Conshohocken, PA: Infinity, 2005), p. 79.

## Sources

- Gerard, L. J. V.,
*The Elements of Geometry, in Eight Books; or, First Step in Applied Logic*(London, Longmans, Green, Reader and Dyer, 1874), p. 285.

- Legendre, A. M.,
*Elements of Geometry and Trigonometry*, Charles Davies, ed. (New York: A. S. Barnes & Co., 1858), p. 119.