circles

A circle is the set of all points in a plane at a fixed distance from a given point. That fixed distance is called the radius, and the point is the center. The distance from the center to any point on the circle is constant. The boundary of the circle is the circle itself; the interior together with the boundary is called the disk.

Key measures include the circumference and the area. The circumference C is 2πr (equivalently πD, where D is the diameter). The area A is πr^2. The ratio of the circumference to the diameter is π. Circles have infinite lines of symmetry and are invariant under rotations about the center.

Common features and related terms include the radius, the diameter (twice the radius), a tangent (a line that touches the circle at exactly one point), a chord (a line segment with endpoints on the circle), an arc (a portion of the circle), and a sector (a region bounded by two radii and an arc). The unit circle is the circle with radius 1, often centered at the origin.

Equations for circles are used to describe them algebraically. In Cartesian coordinates, a circle with center (h,k) and radius r satisfies (x − h)^2 + (y − k)^2 = r^2. The unit circle centered at the origin has x^2 + y^2 = 1. In polar coordinates, a circle can be described by r = constant or by related forms depending on the center.

Circles appear across mathematics, science, engineering, and design, serving as a fundamental reference shape and a model for many natural and applied phenomena.