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Portion of a disk enclosed by two radii and an arcNot to be confused with circular section. The minor sector is shaded in green while the major sector is shaded white.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.1 In the diagram, θ is the central angle, r displaystyle r the radius of the circle, and L displaystyle 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.2TypesA 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.Compass An 8-point windroseTraditionally 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 18th of the circle.Most commonly, octants are seen on the compass rose.AreaSee also: Circular arc § Sector areaThe total area of a circle is πr2. The area of the sector can be obtained by multiplying the circles 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): A = π r 2 θ 2 π = r 2 θ 2 displaystyle A=pi r^2,frac theta 2pi =frac r^2theta 2 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. A = π r 2 L 2 π r = r L 2 displaystyle A=pi r^2,frac L2pi r=frac rL2 Another approach is to consider this area as the result of the following integral: A = ∫ 0 θ ∫ 0 r d S = ∫ 0 θ ∫ 0 r r ~ d r ~ d θ ~ = ∫ 0 θ 1 2 r 2 d θ ~ = r 2 θ 2 displaystyle A=int _0^theta int _0^rdS=int _0^theta int _0^rtilde r,dtilde r,dtilde theta =int _0^theta frac 12r^2,dtilde theta =frac r^2theta 2 Converting the central angle into degrees gives3 A = π r 2 θ ∘ 360 ∘ displaystyle A=pi r^2frac theta ^circ 360^circ PerimeterThe length of the perimeter of a sector is the sum of the arc length and the two radii: P = L + 2 r = θ r + 2 r = r ( θ + 2 ) displaystyle P=L+2r=theta r+2r=r(theta +2) where θ is in radians.Arc lengthThe formula for the length of an arc is:4 L = r θ displaystyle L=rtheta 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.5If the value of angle is given in degrees, then we can also use the following formula by:3 L = 2 π r θ 360 displaystyle L=2pi rfrac theta 360 Chord lengthThe length of a chord formed with the extremal points of the arc is given by C = 2 R sin ⁡ θ 2 displaystyle C=2Rsin frac theta 2 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 alsoCircular 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 sectionEarth quadrantReferences^ Dewan, Rajesh K. (2016). Saraswati Mathematics. New Delhi: New Saraswati House India Pvt Ltd. p. 234. ISBN 978-.^ Achatz, Thomas; Anderson, John G. (2005). Technical shop mathematics. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376. ISBN 978-. 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.: BrooksCole. 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.SourcesGerard, 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.Retrieved from https:en.wikipedia.orgwindex.php?title=Circular_sector&oldid=
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