A figure with a curved boundary made up of all points a fixed distance from a center point is called a circle. That fixed distance is the radius, while the central point is the center.
In Euclidean geometry, every point on a circle is equally far from its center. If the center is at (a, b) and the radius is r, the circle can be represented by (x − a)² + (y − b)² = r². The related disk includes all the points inside the circle as well as its boundary, so “circle” and “disk” are not technically interchangeable.
The circle has been studied since antiquity. Ancient Greek mathematicians explored its symmetry, circumference, area, and relationship to pi. Its circumference is 2πr, and its area is πr². A sphere is the three-dimensional counterpart: it consists of points at a fixed distance from a center in space.
The definition depends on the geometry being used. In ordinary Euclidean geometry the familiar shape is round, but alternative distance rules can produce differently shaped “circles,” including diamond-like forms in taxicab geometry.