# Equation of a circle passing through 3 given points

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

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#### Anton

Created: 2019-02-22 09:10:01, Last updated: 2021-09-30 12:30:30

#### Third point

Digits after the decimal point: 2

#### Center

x

y

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 the 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 a 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 a circle through them.
The coordinates of a center of a circle and it's radius related to the solution like this
$x_c=-a\\y_c=-b\\R=\sqrt{x_c^2+y_c^2-c}$

Knowing center and radius, we can get the equations using Equation of a circle calculator

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PLANETCALC, Equation of a circle passing through 3 given points