A circular sector, also known as circle sector or disk sector or simply a sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, with the smaller area being known as the minor sector and the larger being the major sector.[1] In the diagram, θ is the central angle, the radius of the circle, and is the arc length of the minor sector.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/da/Circle_arc.svg/300px-Circle_arc.svg.png)
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]
Types
editA 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. The arc of a quadrant (a circular arc) can also be termed a quadrant.
Compass
editTraditionally wind directions on the compass rose 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 wind vane typically does not have enough accuracy to allow more precise indication.
The name of the instrument "octant" comes from the fact that it is based on 1/8th of the circle. Most commonly, octants are seen on the compass rose.
Area
editThe 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:
Perimeter
editThe 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
editThe formula for the length of an arc is:[4] 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]
If the value of angle is given in degrees, then we can also use the following formula by:[3]
Chord length
editThe 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
- Earth quadrant
- Sector of (mathematics)
- Spherical sector – the analogous 3D figure
References
edit- ^ Dewan, Rajesh K. (2016). Saraswati Mathematics. New Delhi: New Saraswati House India Pvt Ltd. p. 234. ISBN 978-8173358371.
- ^ Achatz, Thomas; Anderson, John G. (2005). Technical shop mathematics. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376. ISBN 978-0831130862. OCLC 56559272.
- ^ a b Uppal, Shveta (2019). Mathematics: Textbook for class X. New Delhi: National Council of Educational Research and Training. pp. 226, 227. ISBN 978-81-7450-634-4. OCLC 1145113954.
- ^ Larson, Ron; Edwards, Bruce H. (2002). Calculus I with Precalculus (3rd ed.). Boston, MA.: Brooks/Cole. p. 570. ISBN 978-0-8400-6833-0. OCLC 706621772.
- ^ Wicks, Alan (2004). Mathematics Standard Level for the International Baccalaureate : a text for the new syllabus. West Conshohocken, PA: Infinity Publishing.com. p. 79. ISBN 0-7414-2141-0. OCLC 58869667.
Sources
edit- 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.