Circular sector
A circular sector, also known as circle sector or disk sector or simply a sector (symbol: ⌔), is the portion of a
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
A sector with the central angle of 180° is called a
Compass
Traditionally
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
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:
Converting the central angle into degrees gives[3]
Perimeter
The length of the perimeter of a sector is the sum of the arc length and the two radii:
Arc length
The formula for the length of an arc is:[4]
If the value of angle is given in degrees, then we can also use the following formula by:[3]
Chord length
The length of a chord formed with the extremal points of the arc is given by
See also
- 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
- ISBN 978-8173358371.
- OCLC 56559272.
- ^ OCLC 1145113954.
- OCLC 706621772.
- OCLC 58869667.
Sources
- 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.