Mathematics, Statistics & Geometry

Circle Equation Calculator

Generate the standard and general form equations for any circle from its center and radius. See full step-by-step mathematical expansion.

Standard Form
(x - 2)² + (y + 3)² = 25
General Formx² + y² - 4x + 6y - 12 = 0
Diameter10
Area78.54
Calculation StepsCenter (h, k) = (2, -3), Radius (r) = 5 Standard Form: (x - h)² + (y - k)² = r² Standard Form: (x - 2)² + (y + 3)² = 25 General Form: Expand (x² - 2hx + h²) + (y² - 2ky + k²) = r² General Form: x² + y² - 2(2)x - 2(-3)y + (2² + -3² - 5²) = 0 General Form: x² + y² - 4x + 6y - 12 = 0

Calculated locally in your browser. Fast, secure, and private.

Mapping Circles on the Cartesian Plane

The Circle Equation Calculator is the ultimate tool for coordinate geometry. Whether you are plotting objects in a 2D video game or solving high school algebra problems, this calculator instantly translates the physical properties of a circle (center and radius) into its mathematical formulas.

(x−h)2+(y−k)2=r2\begin{aligned} (x - h)^2 + (y - k)^2 = r^2 \end{aligned}

Where:
h=
The x-coordinate of the circle's center
k=
The y-coordinate of the circle's center
r=
The distance from the center to the edge

Two Forms, One Shape

Our calculator generates both widely used mathematical forms of a circle:

  1. Standard Form: Best for human readability and graphing. By looking at it, you can instantly pinpoint the center coordinates and the radius.
  2. General Form: Best for algebraic manipulation, intersection algorithms, and advanced calculus. It removes all parentheses.

How the Expansion Works

Converting from Standard to General form requires expanding the binomials:

  1. Expand: (x - h)² -> x² - 2hx + h²
  2. Expand: (y - k)² -> y² - 2ky + k²
  3. Combine constants: h² + k² - r² becomes the single constant at the end of the general equation. Our calculator provides a full step-by-step breakdown of this exact process.

Applications

  • Computer Graphics: Writing collision detection algorithms using circle boundaries.
  • CNC Machining: Programming toolpaths for circular cuts.
  • Civil Engineering: Designing circular roundabouts or curving road segments on a geographic grid.

Frequently Asked Questions

The standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form makes it very easy to graph the circle.

The general form is x² + y² + Dx + Ey + F = 0. It is obtained by expanding the standard form.

You must use the mathematical technique of 'completing the square' for both the x and y terms to convert it back into standard form.

If the center is at (0,0), both h and k are zero, and the equation simplifies beautifully to x² + y² = r².

No. A radius is a distance, so it must be a positive number. If the right side of the standard equation is 0, it represents a single point. If it is negative, the circle does not exist on the real plane.

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