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. However, differences in worldview (beliefs and culture) had a great impact on artists’ perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits. If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.

Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings. [2] Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi. The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality).

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If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠ BOA and ∠ BPA are supplementary. The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are: Disc: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc, while in everyday the terms "circle" and "disc" may be used interchangeably.

Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle. A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon. 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. Usually, the radius is denoted r {\displaystyle r} and required to be a positive number. A circle with r = 0 {\displaystyle r=0} is a degenerate case consisting of a single point. Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre. Construction through three noncollinear points The word circle derives from the Greek κίρκος/κύκλος ( kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος ( krikos), meaning "hoop" or "ring". [1] The origins of the words circus and circuit are closely related.

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The series of regular polygons with n sides has the circle as its limit as n approaches infinity. This fact was applied by Archimedes to approximate π.