Circle Equation Calculator

Results:

Understanding Circle Equations

Basic Concepts

A circle is a set of points in a plane that are equidistant from a fixed point called the center.

Standard Form: (x-h)² + (y-k)² = r²

General Form: x² + y² + Dx + Ey + F = 0

Center: (-D/2, -E/2)

Radius: r = √((D²+E²)/4 - F)

Properties and Applications

Circle Properties

  • Area = πr²
  • Circumference = 2πr
  • Diameter = 2r
  • Arc length = rθ
  • Sector area = ½r²θ
  • Inscribed angle theorem
  • Power of a point
  • Tangent properties

Applications

  • Engineering design
  • Architectural planning
  • Satellite orbits
  • GPS positioning
  • Computer graphics
  • Radar systems
  • Circular motion
  • Geometric constructions

Advanced Topics

Analytical Geometry

  • Circle-line intersection
  • Circle-circle intersection
  • Tangent lines
  • Polar form
  • Parametric equations
  • Complex plane representation
  • Inversion in circles
  • Circle bundles

Mathematical Properties

Parametric: x = h + r cos(t), y = k + r sin(t)

Polar: r = 2a cos(θ) or r = 2a sin(θ)

Complex: |z - c| = r

Matrix form: [x y 1][A B/2 D/2][x] = 0

Geometric Relationships

  • Apollonian circles
  • Circle of Apollonius
  • Nine-point circle
  • Radical axis
  • Power center
  • Orthogonal circles
  • Cyclic quadrilaterals
  • Circle packing

Applications in Physics

Centripetal force: F = mv²/r

Angular velocity: ω = v/r

Period: T = 2πr/v

Moment of inertia: I = mr²