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 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
Circle
Circle-withsegments.svg
A circle
circumference C
diameter D
radius R
center or origin O
Type
Conic section
Symmetry group
O(2)
Area
πR2
Perimeter
C = 2πR
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.
Contents
Euclid's definition
A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.
— Euclid, Elements, Book I[1]: 4
Topological definition
In the field of topology, a circle is not limited to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy).[2]
Terminology
Annulus: a ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle.
Centre: the point equidistant from all points on the circle.
Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
Circumference: the length of one circuit along the circle, or the distance around the circle.
Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
Disc: the region of the plane bounded by a circle.
Lens: the region common to (the intersection of) two overlapping discs.
Passant: a coplanar straight line that has no point in common with the circle.
Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter.
Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to.
Secant: an extended chord, a coplanar straight line, intersecting a circle in two points.
Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").
All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries.
Chord, secant, tangent, radius, and diameter
Arc, sector, and segment
History
The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo.
The word circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring".[3] The origins of the words circus and circuit are closely related.
Circular piece of silk with Mongol images
Circles in an old Arabic astronomical drawing.
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.
Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[4][5]
