A circular sector or circle sector, is the portion of a circle enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. Its area can be calculated as described below.

Let θ be the central angle, in radians, and ${\displaystyle r}$ the radius. The total area of a circle is ${\displaystyle \pi r^2}$ . The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle and ${\displaystyle 2\pi}$ (because the area of the sector is proportional to the angle, and ${\displaystyle 2\pi}$ is the angle for the whole circle):

${\displaystyle A=\pi r^2\cdot\frac{\theta}{2\pi}=\left(\frac{\theta}{2}\right)r^2=\frac12r^2\theta}$

Also, if ${\displaystyle \theta}$ refers to the central angle in degrees, a similar formula can be derived.

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

Sectors can have special relationships, which include halves, quadrants, and octants.

The length ${\displaystyle L}$ of the arc of a sector is given by the following formula:

${\displaystyle L=\pi\cdot\frac{\theta}{180}r}$

where ${\displaystyle \theta}$ is in degrees.

The length of the perimeter of a sector is sum of arc length and the two radii. It is given by the following formula:

${\displaystyle L=\left(2+\pi\cdot\frac{\theta}{180}\right)r}$

where ${\displaystyle \theta}$ is in degrees.

The sector area of a Polygon is:

${\displaystyle \frac{\theta}{180(n-2)}A_n}$

Circumscribed sector area

The sector area of a circumscribed circle is:

${\displaystyle \frac{\theta}{360}\frac{\pi}{4\sin^2\left(\frac{180}{n}\right)}s^2}$

Inscribed sector area

The sector area of a inscribed circle is:

${\displaystyle \frac{\theta}{360}\frac{\pi}{4\tan^2\left(\frac{180}{n}\right)}s^2}$

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

External links

• Definition and properties of a circle sector With interactive animation
• Wolfram MathWorld: Circular Sector

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