# The Equation of a Circle Through Three Points

You learn in high school algebra that the four conic sections are the hyperbola, parabola, ellipse, and circle. The circle is actually a special kind of ellipse, one in which the major and minor axes have the same length. A circle is defined as a closed curve such that every point on the circumference is equidistant from the center. The distance from the center of the circle to the outer edge is called the radius. In the xy-coordinate plane, the general equation of a circle is

r^{2} = (x - h)^{2} + (y - k)^{2},

where r is the radius, and the point (h, k) is the center of the circle. That is, h is the x-coordinate of the center, and k is the y-coordinate of the center. If you know the coordinates of three points in the xy-plane, you can find the equation of the circle that passes through these points (assuming that the points do not lie on the same line). The process involves setting up three equations in the variables r, h, and k.

You can use the example below as a guide when you do the calculations, or you can use the equation solver on the left. To use the Circle Equation Calculator, just input the coordinates of three points; the calculator will output the equation of the circle that passes between them, or tell you if the points lie on the same line.

## Example Problem

Suppose you are given the three points (1,1), (1,7), and (4,4) and you want to find the equation of the circle that passes through the points. The first step is to set up these 3 equations by plugging the x- and y-coordinates of the points into the circle formula:(1 - h)

^{2}+ (1 - k)

^{2}= r

^{2}

(1 - h)

^{2}+ (7 - k)

^{2}= r

^{2}

(4 - h)

^{2}+ (4 - k)

^{2}= r

^{2}

Notice that the right hand sides are all equal to r

^{2}. This means you can set the left hand sides equal to each other. If you do this for the first and second, and the second and third, you get

(1 - h)

^{2}+ (1 - k)

^{2}= (1 - h)

^{2}+ (7 - k)

^{2}, and

(1 - h)

^{2}+ (7 - k)

^{2}= (4 - h)

^{2}+ (4 - k)

^{2}

The first equation can be simplified:

(1 - h)

^{2}+ (1 - k)

^{2}= (1 - h)

^{2}+ (7 - k)

^{2}

1 - 2h + h

^{2}+ 1 - 2k + k

^{2}= 1 - 2h + h

^{2}+ 49 - 14k + k

^{2}

1 - 2k = 49 - 14k

12k = 48

k = 4

The second equation can also be simplified:

(1 - h)

^{2}+ (7 - k)

^{2}= (4 - h)

^{2}+ (4 - k)

^{2}

1 - 2h + h

^{2}+ 49 - 14k + k

^{2}= 16 - 8h + h

^{2}+ 16 - 8k + k

^{2}

1 - 2h + 49 - 14k = 16 - 8h + 16 - 8k

18 = -6h + 6k

Since k = 4, you can plug this in to solve for h: 18 = -6h + 24. This gives you h = 1.

Now to solve for r. Since you have values for k and h, you can plug these into one of the original equations to find r

^{2}. If you plug k = 4 and h = 1 into the first equation, you get

(1 - 1)

^{2}+ (1 - 4)

^{2}= r

^{2}

0 + 3

^{2}= r

^{2}

9 = r

^{2}

3 = r

So the equation of the circle is (x-1)

^{2}+ (y-4)

^{2}= 3

^{2}. This is a circle centered at the point (1,4) with a radius of 3. You can use this method to find the equation of a circle through any set of three points, or you can use the calculator above.

### Try it!

The applet below shows four points where each subset of three points defines a circle, for a total of four distinct circles. Use your cursor to drag the points around the window to see how the circles change according to the positions of the points

© *Had2Know 2010
*

Applet created with GeoGebra.