Circular sector

Not to be confused with circular section.

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.

The minor sector is shaded in green while the major sector is shaded white.

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

See also: Circular arc § Sector 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): $${\displaystyle A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}}$$

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. $${\displaystyle A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}}$$

Another approach is to consider this area as the result of the following integral: $${\displaystyle A=\int _{0}^{\theta }\int _{0}^{r}dS=\int _{0}^{\theta }\int _{0}^{r}{\tilde {r}}\,d{\tilde {r}}\,d{\tilde {\theta }}=\int _{0}^{\theta }{\frac {1}{2}}r^{2}\,d{\tilde {\theta }}={\frac {r^{2}\theta }{2}}}$$

Converting the central angle into degrees gives^[2] $${\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}}$$

Perimeter

The length of the perimeter of a sector is the sum of the arc length and the two radii: $${\displaystyle P=L+2r=\theta r+2r=r(\theta +2)}$$ where θ is in radians.

Arc length

The formula for the length of an arc is:^[3] $${\displaystyle L=r\theta }$$ 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] $${\displaystyle L=2\pi r{\frac {\theta }{360}}}$$

Chord length

The length of a chord formed with the extremal points of the arc is given by $${\displaystyle C=2R\sin {\frac {\theta }{2}}}$$ where C represents the chord length, R represents the radius of the circle, and θ 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.
• Conic section
• Earth quadrant
• Hyperbolic sector
• Sector of (mathematics)
• Spherical sector – the analogous 3D figure
• Spherical wedge – another 3D generalization