How do you find the centre of a circle from its general equation?

The general equation of a circle is x² + y² + Dx + Ey + F = 0. The centre is at (h, k) = (-D/2, -E/2), and the radius is r = sqrt((D/2)² + (E/2)² - F). For example, x² + y² - 6x + 4y - 12 = 0 gives D = -6, E = 4, F = -12, so the centre is (3, -2) and the radius is sqrt(9 + 4 + 12) = 5.

How do you find the centre of a circle from three points?

Three points on a circle uniquely determine the circle. Substituting each point (x, y) into x² + y² + Dx + Ey + F = 0 gives three linear equations in D, E, and F. Solving this system (using Cramer's rule or row reduction) yields D, E, and F, from which the centre (-D/2, -E/2) and radius sqrt((D/2)² + (E/2)² - F) can be found.

What is the standard form of a circle equation?

The standard (centre-radius) form is (x - h)² + (y - k)² = r², where (h, k) is the centre and r is the radius. The general form x² + y² + Dx + Ey + F = 0 can be converted to standard form by completing the square for both x and y. The two forms are related by h = -D/2, k = -E/2, and r² = h² + k² - F.

Centre of a Circle Calculator

This calculator finds the centre and radius of a circle two ways: from the general equation x² + y² + Dx + Ey + F = 0, or from three points on the circle's circumference. Switch between the two methods using the tabs above the form. For the equation method, enter the coefficients D, E and F. For the three-points method, enter the x and y coordinates of any three points, and the calculator solves the underlying system with Cramer's rule to work out D, E and F for you. Either way you get back the centre as coordinates (h, k), the radius, and the standard form equation, plus a step-by-step working panel showing whether the answer came from completing the square or from solving three linear equations. A summary panel also lists h, k and the radius separately so you can carry the numbers into further calculations or a graph. Use it to check homework, verify a circle equation before plotting it, or work out a circle's equation from three points measured off a diagram. If the three points are collinear, or the equation you enter does not describe a real circle, the calculator flags this instead of returning a false result.

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Verified 2025 Standard Euclidean geometry. Results match completing-the-square and Cramer's rule.

Method

Enter coefficients for x² + y² + Dx + Ey + F = 0

D (coefficient of x)

E (coefficient of y)

F (constant)

Circle Properties

Please check your inputs. The values you entered do not form a valid circle (radius is imaginary or the points are collinear).

Centre

(3, -2)

Radius

Standard Form

(x − 3)2 + (y + 2)2 = 25

Working

Summary

Centre (h, k)
(3, -2)
x and y of centre

h (x-coordinate)

Horizontal position

k (y-coordinate)

-2

Vertical position

Radius

Distance from centre

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:

Group x terms: x² + Dx = (x + D/2)² − (D/2)²
Group y terms: y² + Ey = (y + E/2)² − (E/2)²
Rearrange: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F
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:

Step	Action
1	Write three equations: x_i² + y_i² + Dx_i + Ey_i + F = 0
2	Let R_i = −(x_i² + y_i²); solve the 3x3 linear system [x_i, y_i, 1] [D, E, F] = [R_i]
3	Compute centre: h = −D/2, k = −E/2
4	Compute radius: r = √(h² + k² − F)

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.