In an *x*-*y* coordinate system, the circle with centre (*x*_{0},*y*_{0}) and radius *r* is the set of all points (*x*,*y*) such that

- (
*x*-*x*_{0})^{2}+ (*y*-*y*_{0})^{2}=*r*^{2}.

*x*^{2}+*y*^{2}=*r*^{2}.

All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2&pi and π, respectively. In other words:

- Length of a circle's circumference = 2 × π × radius
- Area of a circle = π × (radius)
^{2}

A line cutting a circle in two places is called a secant, and a line touching the circle in one place is called a tangent. The tangent lines are necessarily perpendicular to the radii, segments connecting the centre to a point on the circle, whose length matches the definition given above. The segment of a secant bound by the circle is called a chord, and the longest chords are those that pass through the centre, called diameters and divided into two radii. The part of a circle cut off by a chord is called a circle segment.

It is possible (Circle points segments proof) to find the maximum number of unique segments generated by running chords between a number of points on the perimeter of a circle.

If only (part of) a circle is known, then the circle's center can be constructed as follows: take two chords, construct perpendicular lines on their midpoints, and find the intersection point of those lines.

A part of a circle bound by two radii is called an arc, and the ratio between the length of an arc and the radius defines the angle between the two radii in radians.

Every triangle gives rise to several circles: its circumcircle containing all three vertices, its incircle lying inside the circle and touching all three sides, the three excircles lying outside the triangle and touching one side and the extensions of the other two, and its nine point circle which contains various important points of the triangle. Thales' theorem states that if the three vertices of a triangle lie on a given circle with one side of the triangle being a diameter of the circle, then the angle opposite to that side is a right angle.

Given any three points which do not lie on a line, there exists precisely one circle containing those points (namely the circumcircle of the triangle defined by the points).

A circle is a kind of conic section, with eccentricity zero. In affine geometry all circles and ellipses become (affinely) isomorphic, and in projective geometry the other conic sections join them. In topology all simple closed curves are homeomorphic to circles, and the word circle is often applied to them as a result. The 3-dimensional analog of the circle is the sphere.

Squaring the circle refers to the (impossible) task of constructing, for a given circle, a square of equal area with ruler and compass alone. Tarski's circle-squaring problem, by contrast, is the task of dividing a given circle into finitely many pieces and reassembling those pieces to obtain a square of equal area. Assuming the axiom of choice, this is indeed possible.

Three-dimensional shapes whose cross-sections in some planes are circles include spheres, spheroids, cylinders, and cones.

See also:

simple:circle