# 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 $(x-h{)}^{2}+(y-k{)}^{2}={r}^{2}$, where $(h,k)$ represents the center and $r$ is the radius.
**General Form:** ${x}^{2}+{y}^{2}+2gx+2fy+c=0$ represents a circle equation with center $(-g,-f)$ 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.