Equation of a Circle Calculator

A circle is defined as the set of all points in a plane that are equidistant from a fixed point known as the center. The standard form of a circle's equation is written as (x - h)² + (y - k)² = r², where (h, k) represents the coordinates of the center and r denotes the radius. When expanded, this transforms into the general form: x² + y² + Dx + Ey + F = 0. Our calculator relates these coefficients using the formulas D = -2h, E = -2k, and F = h² + k² - r². Additionally, it computes key measurements including circumference (C = 2πr) and area (A = πr²).

Let us find the key properties and general form of a circle with a center at (-3, -5) and a radius of 5 units. First, substitute the center (h = -3, k = -5) and radius (r = 5) into the standard form: (x - (-3))² + (y - (-5))² = 5², which simplifies to (x + 3)² + (y + 5)² = 25. Next, expand the binomials: x² + 6x + 9 + y² + 10y + 25 = 25. Subtracting 25 from both sides yields the general equation: x² + y² + 6x + 10y + 9 = 0. Here, the D coefficient is 6, E is 10, and F is 9. The diameter is 10 units, the circumference is approximately 31.42 units, and the area is roughly 78.54 square units.

When converting from the general form to the standard form, remember to use the technique of completing the square for both the x and y terms. Always verify that the coefficient of both x² and y² is 1 before attempting to find the center and radius. Pay close attention to negative signs inside standard form parentheses, as (x - h) means a positive center coordinate h when the sign is negative.