Centre of a Circle Calculator

What Is the Centre of a Circle?

The centre of a circle is the fixed point that is equidistant from every point on the circle. This distance is the radius. In a coordinate system, the centre is written as (h, k), so a circle with centre (h, k) and radius r satisfies the standard equation:

(x − h)² + (y − k)² = r²

Method 1: From the General Equation

Any circle can be written in general form as x² + y² + Dx + Ey + F = 0. To find the centre and radius, complete the square for both x and y:

1. Group x terms: x² + Dx = (x + D/2)² − (D/2)²
2. Group y terms: y² + Ey = (y + E/2)² − (E/2)²
3. Rearrange: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F
4. So h = −D/2, k = −E/2, and r = √((D/2)² + (E/2)² − F)

For a valid circle, the expression under the square root must be positive.

Method 2: From Three Points

Three non-collinear points on a circle uniquely determine that circle. Substituting each point (x_i, y_i) into the general equation gives three linear equations in D, E, and F. This system is solved using Cramer's rule:

Worked Example (Default Values)

General equation: x² + y² − 6x + 4y − 12 = 0, so D = −6, E = 4, F = −12.

• h = −D/2 = −(−6)/2 = 3
• k = −E/2 = −4/2 = −2
• r² = (D/2)² + (E/2)² − F = 9 + 4 + 12 = 25, so r = 5
• Centre: (3, −2), Radius: 5
• Standard form: (x − 3)² + (y + 2)² = 25

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Sources and method: Euclidean geometry standard form of a circle. Completing the square as per any secondary-school coordinate geometry curriculum. Cramer's rule for 3x3 linear systems (standard linear algebra).

Results are exact for the inputs given. Floating-point rounding may cause very small differences in the last decimal place for three-point calculations involving large coordinates.