Circular sector

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
The minor sector is shaded in green while the major sector is shaded white.

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


The total area of a circle is πr2. 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 πr2 by the ratio of L to the total perimeter 2πr.

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

Converting the central angle into degrees gives[3]


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[edit]

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[edit]

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[edit]

  • 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


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