CIRCLE

A circle is the set of all points in a plane that are at a fixed distance, called the radius, from a fixed point, the center. The diameter is twice the radius and passes through the center.

The circumference, or perimeter, of a circle is 2πr, and its area is πr^2. The unit of the area is square units, and the unit circle has radius 1.

Key geometric features include chords, tangents, and arcs. A chord connects two points on the circle; a tangent touches the circle at exactly one point and is perpendicular to the radius at that point. An arc is a continuous portion of the circle. Central angles subtend arcs, and inscribed angles subtend the same arc and are equal.

In Cartesian coordinates, a circle with center (h,k) and radius r is described by (x−h)^2+(y−k)^2=r^2. The unit circle is the special case with h=0, k=0, r=1 and is fundamental in trigonometry.

In three dimensions, a circle is the intersection of a plane with a sphere. The circle term also appears in contexts such as circumcircles of polygons and incircles of triangles.