Circular Sector
**CIRCULAR SECTOR**

*Balmoral Software*

Let a circular sector S
of radius 1 be oriented with the origin at the midpoint of the chord connecting
the endpoints of its arc, and with the center of the circle on the negative
y-axis, so that S is symmetric with respect to that axis. If
β < π/2 is one-half the angle subtended by the sector arc,
then the extreme abscissa points of S are (±sin(β),0), the minimum
ordinate point is (0,-cos(β)), the maximum ordinate point is
(0,1 - cos(β)), and the width x height of the bounding
rectangle of S is 2sin(β) x 1.

Let t represent the usual parameter, ranging from 0 to 2π through the
quadrants in counterclockwise order. In Quadrants I and II, the parametric
equations of the sector arc are:
x(φ) = cos(φ)
y(φ) = sin(φ) - cos(β),

where the angle φ is measured from the center of the circle and ranges from
π/2 - β to π/2 + β. We can scale t to
the angular parameter φ as follows:
Range of t | Range of φ |

[0,π] | [π/2 - β,π/2 + β] |

or equivalently,
φ = 2βt/π + π/2 - β

In Quadrants III and IV, S consists of symmetric line segments joining the
points (-sin(β),0), (0,-cos(β)) and (sin(β),0), so the parametric
functions are linear in t. Combining definitions, we have

### Metrics

The perimeter of the sector is the length of the circular arc (in radians) plus
its two radii:
L = 2β + 2

The area A of S is
β radians and its centroid ordinate is

*Specific Example*

In the remainder of this paper, we use

β = 35° = 7π/36
L = 3.221731

A = 0.610865

d = -0.193180

Bounding rectangle: 1.147153 x 1

### Incircle

Using z = 1 - cos(β) in Lemma C,
we have
R = |d - z| = 0.364505

For verification, we have

### Inellipse

Using z = 1 - cos(β) in Lemma E,
d/dt x(t)[y(t) - z] = d/dt (2t/π - 3)sin(β)[|2t/π - 3|cos(β) - 1]

has a zero in [π,2π) when

The corresponding coordinates are

We then have

For verification,

### Circumcircle

A circumcircle passing through the three corners of S will have a smaller radius
than that of the sector itself, and so will have a greater curvature and contain
the circular arc portion of S. Applying
Lemma TC to an inverted acute
isosceles triangle with h = cos(β) and
w = 2sin(β), the center ordinate (negated) and the radius of
the circumcircle are:
d = h/2 - w^{2}/(8h) = cos(2β)sec(β)/2 = -0.208765
R = h - d = sec(β)/2 = 0.610387

### Circumellipse

Using the same approach as for the circumcircle, we apply
Lemma TE with
h = cos(β) and w = 2sin(β) to find the
parameters of the circumellipse:

### Summary Table

**Figure** | **Parameters** | Perimeter | Area | Centroid |

Incircle | R = 0.364505 | 2.290253 | 0.417404 | (0,-0.183657) |

Inellipse | a = tan(7π/36)/\/3 b = 1/3 | 2.322593 | 0.423345 | (0,-0.152485) |

Circular sector | Width: 2sin(β) Height: 1 | 3.221731 | 0.610865 | (0,-0.193180) |

Circumellipse | a = b = | 3.805114 | 1.136275 | (0,-0.273051) |

Circumcircle | R = sec(β)/2 | 3.835177 | 1.170472 | (0,-0.208765) |

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