# Circle Definition

### Circle Definition:

#### 1. Geometric 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.

#### 2. Key Components:

• 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.

#### 3. Characteristics and Properties:

• 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.

#### 4. Mathematical Representation:

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

#### 5. Properties and Concepts:

• 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 $2\pi r$ (where $r$ is the radius).
• Area: The space enclosed by the circle, calculated as $\pi {r}^{2}$ (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.

#### 6. Applications in Mathematics:

• 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.