Equation of a Circle, General Form (Center anywhere)

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:

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:

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

• Equations of circles and ellipses
• Circles and arcs
• Ellipses
• Coordinate Geometry