Acircular sector, also known ascircle sector ordisk sector or simply asector (symbol:⌔), is the portion of adisk (aclosed region bounded by a circle) enclosed by tworadii and anarc, with the smallerarea being known as theminor sector and the larger being themajor sector.[1] In the diagram,θ is thecentral angle,r the radius of the circle, andL is the arc length of the minor sector.
A sector with the central angle of 180° is called ahalf-disk and is bounded by adiameter and asemicircle.Sectors with other central angles are sometimes given special names, such asquadrants (90°),sextants (60°), andoctants (45°), which come from the sector being one quarter, sixth or eighth part of a full circle, respectively.
The total area of a circle isπr2. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angleθ (expressed in radians) and2π (because the area of the sector is directly proportional to its angle, and2π is the angle for the whole circle, in radians):
The area of a sector in terms ofL can be obtained by multiplying the total areaπr2 by the ratio ofL to the total perimeter2πr.
Another approach is to consider this area as the result of the following integral:
Converting the central angle intodegrees gives[2]
The length of theperimeter of a sector is the sum of the arc length and the two radii:whereθ is in radians.
The formula for the length of an arc is:[3]whereL represents the arc length,r represents the radius of the circle andθ represents the angle in radians made by the arc at the centre of the circle.[4]
If the value of angle is given in degrees, then we can also use the following formula by:[5]
The length of achord formed with the extremal points of the arc is given bywhereC represents the chord length,R represents the radius of the circle, andθ represents the angular width of the sector in radians.
