Convert a Circle Equation to the Standard Form

By Mary Jane Sterling

When the equation of a circle appears in the standard form, it provides you with all you need to know about the circle: its center and radius. With these two bits of information, you can sketch the graph of the circle.

The equation x2 + y2 + 6x – 4y – 3 = 0, for example, is the equation of a circle. You can change this equation to the standard form by completing the square for each of the variables. Just follow these steps:

  1. Change the order of the terms so that the x‘s and y‘s are grouped together and the constant appears on the other side of the equal sign.

    Leave a space after the groupings for the numbers that you need to add:

    x2 + 6x _____ + y2 – 4y _____ = 3

  2. Complete the square for each variable, adding the number that creates perfect square trinomials.

    In the case of the x‘s, you add 9, and with the y‘s, you add 4. Don’t forget to also add 9 and 4 to the right:

    x2 + 6x + 9 + y2 – 4y + 4 = 3 + 9 + 4

    When it’s simplified, you have x2 + 6x + 9 + y2 – 4y + 4 = 16

  3. Factor each perfect square trinomial.

    The standard form for the equation of this circle is (x + 3)2 + (y – 2)2 = 16.

The circle has its center at the point (–3, 2) and has a radius of 4 (the square root of 16). To sketch this circle, you locate the point (–3, 2) and then count 4 units up, down, left, and right; sketch in a circle that includes those points. The figure shows you the way.

With the center, radius, and a compass, you too can sketch this circle.
With the center, radius, and a compass, you too can sketch this circle.