Circular sector
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A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a (a bounded by a circle) enclosed by two and an , where the smaller is known as the ''minor sector'' and the larger being the ''major sector''. In the diagram, θ is the , r the radius of the circle, and L is the arc length of the minor sector. 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.


Types

A sector with the central angle of 180° is called a ' and is bounded by a and a . Sectors with other central angles are sometimes given special names, such as 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 of a quadrant (a ) can also be termed a quadrant.


Compass

Traditionally s on the are given as one of the 8 octants (N, NE, E, SE, S, SW, W, NW) because that is more precise than merely giving one of the 4 quadrants, and the typically does not have enough accuracy to allow more precise indication. The name of the instrument "" comes from the fact that it is based on 1/8th of the circle. Most commonly, octants are seen on the .


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 s gives :A = \pi r^2 \frac


Perimeter

The length of the 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'' (: 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 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

* – 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. * *


References

{{Reflist


Sources

*Gerard, L. J. V., ''The Elements of Geometry, in Eight Books; or, First Step in Applied Logic'' (London, , 1874)
p. 285
*, ''Elements of Geometry and Trigonometry'', , ed. (New York: , 1858)
p. 119
Circles