Circular Sector
A wedge obtained by taking a portion of a disk with central angle 
radians
(
), illustrated above as the shaded region.
A sector of 
radians would be a 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)
|
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(2)
|
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(3)
|
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(4)
|
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(5)
|
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(6)
|
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(7)
|
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(8)
|
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(9)
|
The angle 
obeys the relationships
 |  |  |
(10)
|
 |  |  |
(11)
|
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(12)
|
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(13)
|
The area of the sector is
 |  |  |
(14)
|
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(15)
|
(Beyer 1987). The area can also be found by direct integration as
 |
(16)
|
It follows that the weighted mean of the 
is
 |  |  |
(17)
|
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(18)
|
so the geometric centroid of the circular sector is
 |  |  |
(19)
|
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(20)
|
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(21)
|
(Gearhart and Schulz 1990). Checking shows that this obeys the proper limits 
for a semicircle
(
) and 
for
an isosceles triangle (
).
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.
Weisstein, Eric W. "Circular Sector." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CircularSector.html
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