In Euclidean geometry, a circle is that set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference, which more usually means the length of the circle.
In coordinate geometry a circle with centre (x0,y0) and radius r is the set of all points (x,y) such that
(x - x0)2 + (y - y0)2 = r2
A circle is thus a kind of conic section, with eccentricity zero. All circles are similar, so the ratio between the circumference and radius and that between the area and radius square are both constants. These are 2π and π, respectively, and this is the best known definitions of that constant.
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 chord is that which passes through the centre, called the diameter and divided into two radii.
In affine geometry all circles and ellipses become congruent, 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.