# Circular sector

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Circular sector
A circular sector is shaded in green

A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, r the radius of the circle, and L is the arc length of the minor sector.

A sector with the central angle of 180° is called a semicircle. Sectors with other central angles are sometimes given special names, these include quadrants (90°), sextants (60°) and octants (45°).

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.

## 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 and (because the area of the sector is proportional to the angle, and is the angle for the whole circle):

$A = \pi r^2 \cdot \frac{\theta}{2 \pi} = \frac{r^2 \theta}{2}$

Another approach is to consider this area as the result of the following integral :

$A = \int_0^\theta\int_0^r dS=\int_0^\theta\int_0^r \tilde{r} d\tilde{r} d\tilde{\theta} = \int_0^\theta \frac{1}{2} r^2 d\tilde{\theta} = \frac{r^2 \theta}{2}$

Converting the central angle into degrees gives

$A = \pi r^2 \cdot \frac{\theta ^{\circ}}{360}$

## Perimeter

The length of the perimeter of a sector is the sum of the arc length and the two radii:

$P = L + 2r = \theta r + 2r = r \left( \theta + 2 \right)$

## Center of Mass

The distance from the center of the circle (that the sector is a part of) to the center of mass of the sector is two thirds of the corresponding distance for the center of mass of the arc of the sector. In particular, as the central angle approaches zero the center of mass of the arc is at distance r from the center of the circle, so that of the sector is at distance 2r/3. As the central angle approaches 2π (the whole circle), the center of mass of the arc converges to the center of the circle, whence so does that of the circular sector.

## References

• Gerard, L. J. V. The Elements of Geometry, in Eight Books; or, First Step in Applied Logic, London, Longman's Green, Reader & Dyer, 1874. p. 285

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