Circular Sector


A circular sector is a wedge obtained by taking a portion of a disk with central angle theta<pi radians (180 degrees), illustrated above as the shaded region. A sector with central angle of pi radians would correspond to a filled semicircle. Let R be the radius of the circle, a the chord length, s the arc length, h the sagitta (height of the arced portion), and r the apothem (height of the triangular portion). Then


The angle theta obeys the relationships


The area of the sector is


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


It follows that the weighted mean of the y is


so the geometric centroid of the circular sector is


(Gearhart and Schulz 1990). Checking shows that this obeys the proper limits y^_=4R/(3pi) for a semicircle (theta=pi) and y^_->2R/3 for an isosceles triangle (theta->0).

See also

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

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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 sinx/x." 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.

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Circular Sector

Cite this as:

Weisstein, Eric W. "Circular Sector." From MathWorld--A Wolfram Web Resource.

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