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Equation of a circle passing through 3 given points

This online calculator finds equation of a circle passing through 3 given points

This online calculator finds circle passing through three given points. It outputs center and radius of a circle, circle equations and draws a circle on a graph. The method used to find circle center and radius is described below the calculator.

PLANETCALC, Equation of a circle passing through 3 given points

Equation of a circle passing through 3 given points

First point

Second point

Third point

Digits after the decimal point: 2

Center

x
 
y
 
Radius
 
Equation of a circle in standard form
 
Equation of a circle in general form
 
Parametric equations of a circle
 

How to find a circle passing through 3 given points

Let's recall how equation of a circle looks like in general form:
x^2+y^2+2ax+2by+c=0

Since all three points should belong to one circle, we can write system of equations

x_1^2+y_1^2+2ax_1+2by_1+c=0\\x_2^2+y_2^2+2ax_2+2by_2+c=0\\x_3^2+y_3^2+2ax_3+2by_3+c=0

The values (x_1, y_1), (x_2, y_2) and (x_3, y_3) are known. Let's rearrange with respect to unknowns a, b and c.

2x_1a+2y_1b+c + x_1^2+y_1^2+=0\\2x_2a+2y_2b+c+x_2^2+y_2^2=0\\2x_3a+2y_3b+c+x_3^2+y_3^2=0

Now we have three linear equations for three unknowns - system of linear equations with the following matrix form:
\begin{bmatrix}2x_1 & 2y_1 & 1 \\2x_2 & 2y_2 & 1 \\2x_3 & 2y_3 & 1 \\\end{bmatrix} * \begin{bmatrix}a\\b\\c\\\end{bmatrix} = \begin{bmatrix}-(x_1^2+y_1^2)\\-(x_2^2+y_2^2)\\-(x_3^2+y_3^2)\\\end{bmatrix}

We can solve it using, for example, Gaussian elimination like in Gaussian elimination. No solution means that points are co-linear and it is impossible to draw circle through them.
The coordinates of a center of a circle and it's radius related to the solution like this
x_c=-a\\y_c=-b\\R=\sqrt{x_c^2+y_c^2-c^2}

Knowing center and radius we can get the equations using Equations of a circle with given center and radius in different forms

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