Circle Equation Calculator

Find the equation of a circle from center and radius, or convert between standard and general forms.

Choose Input Method

Result

Standard Form
(x-3)² + (y+2)² = 25
circle equation
Center (h, k)(3, -2)
Radius (r)5
Diameter (d)10
Standard Form(x-3)²+(y+2)²=25
General Formx²+y²-6x+4y-12=0
Area78.5398
Circumference31.4159

Step-by-Step Solution

(x-h)² + (y-k)² = r²

Understanding Circle Equations

The equation of a circle in a coordinate plane describes all points (x, y) that are at a fixed distance (the radius) from a fixed point (the center). There are two primary forms for writing circle equations: the standard form and the general form. Converting between these forms is a fundamental skill in analytic geometry.

The standard form directly reveals the center and radius, making it the more intuitive representation. The general form is a polynomial equation that can be derived by expanding the standard form, and it is the form you typically encounter when working with systems of equations or conic sections.

Forms of the Circle Equation

Standard Form

Directly shows center (h, k) and radius r. Most useful for graphing and geometric analysis.

(x-h)² + (y-k)² = r²

General Form

Expanded polynomial form. Coefficients of x² and y² must be equal.

Ax² + Ay² + Dx + Ey + F = 0

Center at Origin

When the center is (0, 0), the equation simplifies considerably.

x² + y² = r²

Parametric Form

Express x and y as functions of a parameter t (angle).

x = h + r cos t, y = k + r sin t

General to Standard

Convert by completing the square for both x and y terms.

h = -D/(2A), k = -E/(2A)

Radius from General

After dividing by A, the radius is computed from the coefficients.

r = √(h² + k² - F/A)

Converting General Form to Standard Form

To convert from general form Ax² + Ay² + Dx + Ey + F = 0 to standard form, the process involves completing the square. First, divide the entire equation by A (if A is not 1). Then group the x terms and y terms separately, and complete the square for each group.

Step-by-Step Process

  1. Start with x² + y² + (D/A)x + (E/A)y + (F/A) = 0 (after dividing by A).
  2. Group: (x² + (D/A)x) + (y² + (E/A)y) = -(F/A).
  3. Complete the square for x: add (D/(2A))² to both sides.
  4. Complete the square for y: add (E/(2A))² to both sides.
  5. Result: (x + D/(2A))² + (y + E/(2A))² = (D/(2A))² + (E/(2A))² - F/A.
  6. Identify: h = -D/(2A), k = -E/(2A), r² = h² + k² - F/A.

Converting Standard Form to General Form

To convert from standard form to general form, simply expand the squares and rearrange. Starting with (x-h)² + (y-k)² = r², expand to get x² - 2hx + h² + y² - 2ky + k² = r². Rearranging gives x² + y² - 2hx - 2ky + (h² + k² - r²) = 0.

Applications of Circle Equations

Circle equations are used extensively in computer graphics to draw and detect circles, in GPS technology for trilateration (determining position from distances to satellites), in physics for modeling circular motion, and in engineering for designing circular components.

  • Computer Graphics: Circle equations are used for rendering, collision detection, and defining circular boundaries.
  • GPS & Navigation: Trilateration uses circles (in 2D) or spheres (in 3D) to determine position from distance measurements.
  • Physics: Circular orbits, centripetal motion, and wave patterns use circle equations.
  • Engineering: Design specifications for circular components, tolerances, and clearances.
  • Conic Sections: A circle is a special case of an ellipse where both axes are equal, connecting to the broader study of conic sections.

Tips for Using This Calculator

  • For standard form input, enter the center coordinates and radius directly.
  • For general form input, ensure the coefficients of x² and y² are equal (both are A).
  • The radius must be positive; if the general form yields a negative value under the square root, no real circle exists.
  • Negative center coordinates are fully supported.