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300px, 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 Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * ''Disc'' (magazine), a Briti ...
enclosed by two
radii In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin ''radius'', meaning ray but also the spoke of a c ...
and an arc, where the smaller
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of ...
is known as the minor sector and the larger being the major sector. In the diagram, θ is the
central angleAngle AOB is a central angle A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those tw ...

central angle
, 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 half-disk and is bounded by a
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for th ...
and a
semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of sym ...

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.


Area

The total area of a circle is ''r''. 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 2 (because the area of the sector is directly proportional to its angle, and 2 is the angle for the whole circle, in radians): :A = \pi r^2\, \frac = \frac The area of a sector in terms of ''L'' can be obtained by multiplying the total area ''r'' by the ratio of ''L'' to the total perimeter 2''r''. :A = \pi r^2\, \frac = \frac 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\, d\tilde\, d\tilde = \int_0^\theta \frac12 r^2\, d\tilde = \frac Converting the central angle into degrees gives :A = \pi r^2 \frac


Perimeter

The length of the
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical ...
of a sector is the sum of the arc length and the two radii: :P = L + 2r = \theta r + 2r = r (\theta + 2) where ''θ'' is in radians.


Arc length

The formula for the length of an arc is: : L = r \theta 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.Wicks, A., ''Mathematics Standard Level for the International Baccalaureate'' ( West Conshohocken, PA: Infinity, 2005)
p. 79
If the value of angle is given in degrees, then we can also use the following formula by: :L = 2 \pi r \frac


Chord length

The length of a chord formed with the extremal points of the arc is given by :C = 2R\sin\frac 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

{{Reflist


Sources

*Gerard, L. J. V., ''The Elements of Geometry, in Eight Books; or, First Step in Applied Logic'' (London, Longman, Longmans, Green, Reader and Dyer, 1874)
p. 285
*Adrien-Marie Legendre, Legendre, A. M., ''Elements of Geometry and Trigonometry'', Charles Davies (professor), Charles Davies, ed. (New York: Alfred Smith Barnes#A. S. Barnes & Co., A. S. Barnes & Co., 1858)
p. 119
Circles