Circle

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with {\displaystyle r=0}r=0 is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term “circle” may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.

A circle may also be defined as a special kind of ellipse in which the two foci are coincident, the eccentricity is 0, and the semi-major and semi-minor axes are equal; or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

1 Euclid’s definition

2 Topological definition

3 Terminology

4 History

5 Analytic results

5.1 Circumference

5.2 Area enclosed

5.3 Equations

5.4 Tangent lines

6 Properties

6.1 Chord

6.2 Tangent

6.3 Theorems

6.4 Inscribed angles

6.5 Sagitta

7 Compass and straightedge constructions

7.1 Construction with given diameter

7.2 Construction through three noncollinear points

8 Circle of Apollonius

8.1 Cross-ratios

8.2 Generalised circles

9 Inscription in or circumscription about other figures

10 Limiting case of other figures

11 In other p-norms

12 Locus of constant sum

13 Squaring the circle

14 Significance in art and symbolism

15 See also

15.1 Specially named circles