Circle Definition

A circle is a closed curve on a plane. It is a set of all points that are equidistant from a single fixed point called the center.
The equidistant distance from the center to any point on the curve is called the radius of the circle.

Center: The fixed point in the plane from which all points on the circle are equidistant.
Radius: The distance from the center to any point on the circle. All radii of a circle are of equal length.

Symmetry: A circle exhibits radial symmetry, meaning any line passing through its center divides it into two symmetrical halves.
Constant Distance: All points on the circle maintain the same distance from the center.

Coordinate Geometry: The equation of a circle in the Cartesian plane is $(x - h)^{2} + (y - k)^{2} = r^{2}$ , where $(h, k)$ represents the center and $r$ is the radius.
General Form: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ represents a circle equation with center $(- g, - f)$ and radius $\sqrt{g^{2} + f^{2} - c}$ .

Center: The point at the center of the circle from which all points on the circle are equidistant.
Radius: The distance from the center to any point on the circle's circumference.
Diameter: Twice the radius; it is a line passing through the center and two points on the circle's circumference.
Circumference: The perimeter of the circle, calculated as $2 π r$ (where $r$ is the radius).
Area: The space enclosed by the circle, calculated as $π r^{2}$ (where $r$ is the radius).
Chord: A line segment connecting two points on the circle's circumference.
Tangent: A line that intersects the circle at exactly one point, perpendicular to the radius at that point.
Secant: A line that intersects the circle at two distinct points.

Geometry: Used extensively in geometric constructions, theorems, and proofs.
Trigonometry: Circles are fundamental in trigonometric functions and unit circles.
Calculus: Integral calculus involves circles in calculations related to areas and volumes.