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Circular sector

Portion of a disk enclosed by two radii and an arc From Wikipedia, the free encyclopedia

Circular sector
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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.[1] In the diagram, θ is the central angle, r the radius of the circle, and L is the arc length of the minor sector.

Thumb
The minor sector is shaded in green while the major sector is shaded white.
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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 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[2]

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Perimeter

The length of the perimeter of a sector is the sum of the arc length and the two radii: where θ is in radians.

Arc length

The formula for the length of an arc is:[3] where L 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.[4]

If the value of angle is given in degrees, then we can also use the following formula by:[5]

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Chord length

The length of a chord formed with the extremal points of the arc is given by where C represents the chord length, R represents the radius of the circle, and θ represents the angular width of the sector in radians.

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See also

References

Sources

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