A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.: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, 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 arc of a quadrant (a circular arc) 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.: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:
The length of the perimeter of a sector is the sum of the arc length and the two radii:
where θ is in radians.
The formula for the length of an arc is::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.:79
If the value of angle is given in degrees, then we can also use the following formula by:
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.
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- Achatz, T., & Anderson, J. G., with McKenzie, K., ed., Technical Shop Mathematics (New York: Industrial Press, 2005), p. 376.
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- Larson, R., & Edwards, B. H., Calculus I with Precalculus (Boston: Brooks/Cole, 2002), p. 570.
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