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# Circular Sector

A circular sector is a wedge obtained by taking a portion of a disk with central angle radians (), illustrated above as the shaded region. A sector with central angle of radians would correspond to a filled semicircle. Let be the radius of the circle, the chord length, the arc length, the sagitta (height of the arced portion), and the apothem (height of the triangular portion). Then

 (1) (2) (3) (4) (5) (6) (7) (8) (9)

The angle obeys the relationships

 (10) (11) (12) (13)

The area of the sector is

 (14) (15)

(Beyer 1987). The area can also be found by direct integration as

 (16)

It follows that the weighted mean of the is

 (17) (18)

so the geometric centroid of the circular sector is

 (19) (20) (21)

(Gearhart and Schulz 1990). Checking shows that this obeys the proper limits for a semicircle () and for an isosceles triangle ().

Circle-Circle Intersection, Circular Sector Line Picking, Circular Segment, Lens, Obtuse Triangle

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## References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 125, 1987.Gearhart, W. B. and Schulz, H. S. "The Function ." College Math. J. 21, 90-99, 1990.Harris, J. W. and Stocker, H. "Sector." §3.8.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 91-92, 1998.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948.

Circular Sector

## Cite this as:

Weisstein, Eric W. "Circular Sector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CircularSector.html