Equation of a Circle, General Form (Center anywhere)

A circle can be defined as the locus of all points that satisfy the equation
(x-h)2 + (y-k)2 = r2
where r is the radius of the circle,
and h,k are the coordinates of its center.
Try this Drag the point C and note how h and k change in the equation. Drag P and note how the radius squared changes in the equation.

Recall from Basic Equation of a Circle, that when the circle's center is at the origin, the formula is When the circle center is elsewhere, we need a more general form. We add two new variables h and k that are the coordinates of the circle center point:

(x-h)2 + (y-k)2 = r2
We subtract these from x and y in the equation to translate ("move") the center back to the origin.

If you compare the two formulae, you will see that the only difference is that the h and k variables are subtracted from the x and y terms before squaring them:

Basic (x)2    + (y)2    = r2
General (x-h)2 + (y-k)2 = r2

Example

When we see the equation of a circle such as we know it is a circle of radius 9 with its center at   x = 3,   y = –2.

  • The radius is 9 because the formula has r2 on the right side. 9 squared is 81.
  • The y coordinate is negative because the y term in the general equation is (y-k)2.
    In the example, the equation has (y+2), so k must be negative: (y– (–2))2 becomes (y+2)2.

If the circle center is at the origin

The equation is then a little simpler. Since the center is at the origin, h and k are both zero. So the general form becomes which simplifies down to the basic form of the circle equation: For more on this see Basic Equation of a Circle.

Parametric form

Instead of using the Pythagorean Theorem to solve the right triangle in the circle above, we can also solve it using trigonometry. This produces the so-called parametric form of the circle equation as described in Parametric Equation of a Circle. This parametric form is especially useful in computer algorithms that draw circles and ellipses. It is described in An Algorithm for Drawing Circles.

Things to try

  • In the applet above, click 'reset' and 'hide details'.
  • Drag the points C and P to create a new circle.
  • Write the general formula for the resulting circle.
  • Click on 'show details' to check your result.

Related topics

Use the links below for more information on topics related to this page.
COMMON CORE

Math Open Reference now has a Common Core alignment.

See which resources are available on this site for each element of the Common Core standards.

Check it out