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A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a
closed region In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space or the complex coordinate space . This is a different concept ...
bounded by a circle) enclosed by two
radii In classical geometry, a radius ( : radii) 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 ...
and an arc, where the smaller
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
is known as the ''minor sector'' and the larger being the ''major sector''. In the diagram, is the
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 two points, and the arc le ...
, 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 '' 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 Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...
) can also be termed a quadrant.


Compass

Traditionally wind directions on the
compass rose A compass rose, sometimes called a wind rose, rose of the winds or compass star, is a figure on a compass, map, nautical chart, or monument used to display the orientation of the cardinal directions (north, east, south, and west) and thei ...
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 A wind vane, weather vane, or weathercock is an instrument used for showing the direction of the wind. It is typically used as an architectural ornament to the highest point of a building. The word ''vane'' comes from the Old English word , m ...
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 A compass rose, sometimes called a wind rose, rose of the winds or compass star, is a figure on a compass, map, nautical chart, or monument used to display the orientation of the cardinal directions (north, east, south, and west) and thei ...
.


Area

The total area of a circle is . 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 (because the area of the sector is directly proportional to its angle, and 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 \frac 1 2 r^2\, d\tilde = \frac Converting the central angle into degrees gives A = \pi r^2 \frac


Perimeter

The length of the perimeter of a sector is the sum of the arc length and the two radii: P = L + 2r = \theta r + 2r = r (\theta + 2) where is in radians.


Arc length

The formula for the length of an arc is: L = r \theta where 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. 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 represents the chord length, represents the radius of the circle, and represents the angular width of the sector in radians.


See also

*
Circular segment In geometry, a circular segment (symbol: ), also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is ...
– 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 In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
*
Earth quadrant In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is t ...


References

{{Reflist


Sources

*Gerard, L. J. V., ''The Elements of Geometry, in Eight Books; or, First Step in Applied Logic'' (London,
Longmans, Green, Reader and Dyer Longman, also known as Pearson Longman, is a publishing company founded in London, England, in 1724 and is owned by Pearson PLC. Since 1968, Longman has been used primarily as an imprint by Pearson's Schools business. The Longman brand is als ...
, 1874)
p. 285
* Legendre, A. M., ''Elements of Geometry and Trigonometry'', Charles Davies, ed. (New York: A. S. Barnes & Co., 1858)
p. 119
Circles