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
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(1)
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(2)
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(3)
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(4)
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(8)
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The angle 
obeys the relationships
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(10)
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(12)
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The area of the sector is
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(14)
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(Beyer 1987). The area can also be found by direct integration as
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(16)
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It follows that the weighted mean of the 
is
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(17)
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so the geometric centroid of the circular sector is
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(19)
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(20)
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(21)
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(Gearhart and Schulz 1990). Checking shows that this obeys the proper limits 
for a semicircle
(
) and 
for an isosceles
triangle (
).
." College Math. J. 21,
90-99, 1990.Harris, J. W. and Stocker, H. "Sector." §3.8.4
in