A circular segment is a portion of a disk bounded by a chord and a circular arc. The illustration
above shows a minor circular segment with central angle 
radians.
The entire wedge-shaped area is known as a circular
sector.
Circular segments are implemented in the Wolfram Language as DiskSegment[
x, y
, r, 
q1, q2
]. Elliptical segments are similarly implemented as DiskSegment[
x, y
, 
r1, r2
, 
q1, q2
].
For this minor segment, let 
be the radius of the circle,
the chord length,
the arc length,
the height of the arced portion, and
the height of the triangular portion.
Then the radius is
 |
(1)
|
the arc length is
 |
(2)
|
the height 
is
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
and the length of the chord is
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
From elementary trigonometry, the angle 
obeys the relationships
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
The area 
of the (shaded) segment is then simply given by the area
of the circular sector (the entire wedge-shaped
portion) minus the area of the bottom triangular portion,
 |
(14)
|
Plugging in gives
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
where the area formula for an isosceles triangle has been used (Beyer 1987). These formulas find application in the common case of determining the volume of fluid in a cylindrical segment (i.e., horizontal cylindrical tank) based on the height of the fluid in the tank.
With the circle centered at the origin and the chord at 
, the area can also be found directly
by integration as
 |
(19)
|
The integral of 
over this area is
 |  |  |
(20)
|
 |  |  |
(21)
|
so the 
-coordinate
of the geometric centroid of the circular segment
is
 |
(22)
|
This gives 
as 
,
the semicircle value, and 
as 
, when the segment shrinks to the top of the circle.
Finding the value of 
such that the circular segment (left figure) has area equal
to 1/4 of the circle (right figure) is sometimes known
as the quarter-tank problem.
Approximate formulas for the arc length and area are (Harris and Stocker 1998)
 |
(23)
|
accurate to within 0.3% relative error for 
, and
 |
(24)
|
accurate to within 0.15% relative error for 
and 0.8% for 
.
