The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord. [12] :p.71

Sector: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.

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.

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. A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon. [18] Every regular polygon and every triangle is a tangential polygon. The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8 r 2 − 4 p 2, where r is the circle radius, and p is the distance from the centre point to the point of intersection. [11] The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry, and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2, R). The group of rotations alone is the circle group T. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

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 two angles are inscribed on the same chord and on the same side of the chord, then they are equal. 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.