Circle Equation Calculator

Convert circle equations between standard, general, and parametric forms.

Circle Equation Forms

A circle can be represented in three main equation forms, each useful for different purposes.

1. Standard Form

Formula: (x - h)² + (y - k)² = r²

Where (h, k) is the center and r is the radius. This form clearly shows the center and radius, making it ideal for graphing and understanding the circle's position and size.

2. General Form

Formula: x² + y² + Dx + Ey + F = 0

Where:

  • D = -2h
  • E = -2k
  • F = h² + k² - r²

This form is useful for algebraic manipulation and solving systems of equations.

3. Parametric Form

Formulas:

  • x = h + r·cos(t)
  • y = k + r·sin(t)

Where t is a parameter (typically 0 ≤ t ≤ 2π). This form is useful for tracing the circle point by point and in computer graphics applications.

Converting Between Forms

From General to Standard:

  • h = -D/2
  • k = -E/2
  • r = √(h² + k² - F)

Example: A circle with center (3, -2) and radius 4 has:
• Standard form: (x - 3)² + (y + 2)² = 16
• General form: x² + y² - 6x + 4y - 3 = 0
• Parametric form: x = 3 + 4cos(t), y = -2 + 4sin(t)

This calculator provides mathematical conversions between circle equation forms based on standard formulas. Results are for educational and informational purposes. While we strive for accuracy, please verify important calculations independently.

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