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Circular sector

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Title: Circular sector  
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Subject: Area, Angular units, Angle, Track and field, Arc (geometry)
Collection: Circles
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Circular sector

A circular sector is shaded in green

A circular sector or circle sector (symbol: ), 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 half-disk and is bounded by a diameter and 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.

Contents

  • Area 1
  • Arc length 2
  • Perimeter 3
  • See also 4
  • References 5
  • External links 6

Area

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

A = \pi r^2 \cdot \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 \pi r^2by the ratio of L to the total perimeter 2\pi r.

A = \pi r^2 \cdot \frac{L}{2\pi r} = \frac{r \cdot L}{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}

Arc length

The length L of the arc of a sector is given by

L=\theta \cdot r,

where θ is in radians.

If the angle is given in degrees, then

L=\theta ^{\circ} \cdot r \cdot \frac{\pi}{180}

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)

where θ is 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

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

External links

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