Circles - Circles and its Related Terms

A circle is a collection of all points in a plane which are at a constant distance from a fixed point in the same plane. The fixed point is called the center ($O$) and the constant distance is the radius ($r$). Visually, it is a perfectly round loop where every point on the boundary is equidistant from the center.

The circle divides the plane into three parts: the interior (inside the circle), the circle itself (the boundary), and the exterior (outside the circle). The region consisting of the circle and its interior is called the circular region.

A chord is a line segment joining any two points on the circle's circumference. If a chord passes through the center of the circle, it is called the diameter ($d$). The diameter is the longest chord in a circle and is twice the length of the radius ($d = 2r$). Visually, the diameter acts as an axis of symmetry splitting the circle into two identical halves.

A piece of a circle between two points is called an arc. If the distance between two points is less than half the circumference, it forms a minor arc; if it is more than half, it forms a major arc. When the points are ends of a diameter, the arc is called a semicircle.

The length of the complete circular boundary is called its circumference. This is equivalent to the perimeter of the circle. Visually, if you were to cut the circle at one point and straighten it out, the length of that line would be the circumference ($2\pi r$).

A segment is the region between a chord and either of its arcs. The region between the chord and the minor arc is the minor segment (often looking like a small 'cap'), and the region between the chord and the major arc is the major segment.

A sector is the region between an arc and the two radii joining the center to the endpoints of the arc. It resembles a 'slice of pizza'. The minor sector corresponds to the minor arc, and the major sector corresponds to the major arc.

Two or more circles are said to be concentric if they have the same center but different radii. Visually, these look like a target or a 'bullseye' pattern where circles are nested inside one another.