A circle is the set of points equidistant from a given point . The distance from the Center is called the
Radius, and the point is called the Center. Twice the Radius is known as the Diameter
. The Perimeter of a circle is called the Circumference, and is given by

(1) |

The region of intersection of two circles is called a Lens. The region of intersection of three symmetrically placed circles (as in a Venn Diagram), in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux Triangle.

The parametric equations for a circle of Radius are

(2) | |||

(3) |

For a body moving uniformly around the circle,

(4) | |||

(5) |

and

(6) | |||

(7) |

When normalized, the former gives the equation for the unit Tangent Vector of the circle, . The circle can also be parameterized by the rational functions

(8) | |||

(9) |

but an Elliptic Curve cannot. The following plots show a sequence of Normal and Tangent Vectors for the circle.

The Arc Length , Curvature , and Tangential Angle of the circle are

(10) | |||

(11) | |||

(12) |

The Cesàro Equation is

(13) |

In Polar Coordinates, the equation of the circle has a particularly simple form.

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

The equation of a circle passing through the three points for , 2, 3 (the Circumcircle of the
Triangle determined by the points) is

(20) |

(21) |

(22) |

(23) | |||

(24) |

and the Radius as

(25) |

(26) | |||

(27) | |||

(28) | |||

(29) |

Four or more points which lie on a circle are said to be Concyclic. Three points are trivially concyclic since three noncollinear points determine a circle.

The Circumference-to-Diameter ratio for a circle is constant as the size of the circle is changed (as
it must be since scaling a plane figure by a factor increases its Perimeter by ), and also scales by
. This ratio is denoted (Pi), and has been proved Transcendental.
With the Diameter and the Radius,

(30) |

(31) |

Now for a few geometrical derivations. Using concentric strips, we have

As the number of strips increases to infinity, we are left with a Triangle on the right, so

(32) |

As the number of wedges increases to infinity, we are left with a Rectangle, so

(33) |

**References**

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 125 and 197, 1987.

Casey, J. ``The Circle.'' Ch. 3 in
*A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing
an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl.* Dublin: Hodges, Figgis, & Co., pp. 96-150, 1893.

Courant, R. and Robbins, H.
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 74-75, 1996.

Dunham, W. ``Archimedes' Determination of Circular Area.'' Ch. 4 in
*Journey Through Genius: The Great Theorems of Mathematics.*
New York: Wiley, pp. 84-112, 1990.

Eppstein, D. ``Circles and Spheres.'' http://www.ics.uci.edu/~eppstein/junkyard/sphere.html.

Lawrence, J. D. *A Catalog of Special Plane Curves.* New York: Dover, pp. 65-66, 1972.

MacTutor History of Mathematics Archive. ``Circle.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Circle.html.

Pappas, T. ``Infinity & the Circle'' and ``Japanese Calculus.'' *The Joy of Mathematics.*
San Carlos, CA: Wide World Publ./Tetra, pp. 68 and 139, 1989.

Pedoe, D. *Circles: A Mathematical View, rev. ed.* Washington, DC: Math. Assoc. Amer., 1995.

Yates, R. C. ``The Circle.'' *A Handbook on Curves and Their Properties.* Ann Arbor, MI: J. W. Edwards, pp. 21-25, 1952.

© 1996-9

1999-05-26