Circular sector
A circular sector, also known as circle sector or disk sector or simply a sector, is the portion of a disk 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, sextants, and octants, 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 the sector can be obtained by multiplying the circle's area by the ratio of the angle and :The area of a sector in terms of can be obtained by multiplying the total area by the ratio of to the total perimeter.
Another approach is to consider this area as the result of the following integral:
Converting the central angle into degrees gives
Perimeter
The length of the perimeter of a sector is the sum of the arc length and the two radii:where is in radians.
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:
Chord length
The length of a chord formed with the extremal points of the arc is given bywhere represents the chord length, represents the radius of the circle, and represents the angular width of the sector in radians.