Our equation of a circle calculator uses coordinate geometry principles to generate circle equations in both standard and general forms. The calculation is based on the definition that a circle is the set of all points equidistant from a center point, expressed algebraically using the distance formula.

Mathematical Foundation

The circle equation derives from the distance formula. For a point (x, y) to be on a circle with center (h, k) and radius r, the distance from (x, y) to (h, k) must equal r. Using the distance formula: √((x-h)² + (y-k)²) = r. Squaring both sides gives the standard form: (x-h)² + (y-k)² = r². Expanding this algebraically yields the general form.

• Circle is defined as all points at distance r from center (h, k)
• Standard form directly shows center and radius
• General form results from expanding and rearranging standard form
• Completing the square converts general form back to standard form
• Circumference = 2πr, Area = πr², Diameter = 2r
• Circle at origin has equation x² + y² = r² (h = 0, k = 0)

Sources & References