Wikipedia / Circular sector
Wikipedia / Circular sector
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. In the diagram, is the central angle, the radius of the circle, and is the arc length of the minor sector.
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 quarter, sixth or eighth part of a full circle, respectively.
Area
The total area of a circle is . The area of a sector in terms of can be obtained by multiplying the total area by the ratio of to the total perimeter The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle (expressed in radians) and (because the area of the sector is directly proportional to its angle, and
Converting the central angle into degreess gives
Arc length
The formula for the length of an arc is: where represents the arc length, represents the radius of the circle and 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:
Perimeter
The length of the perimeter where is in radians.
Chord length
The length of a chord 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 – 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.
Spherical sector – the analogous 3D figure
Spherical wedge – another 3D generalization