We can use the Pythagorean Theorem to test whether a particular point lies on a particular circle. For example, we know the point lies on the circle centered at with radius 5 because . We can use this reasoning to test whether a general point lies on a circle with center and radius . This is the general equation for a circle: .

Similarly, if we know the equation for a circle, then we can use that to find the center and radius. For example, a circle with equation has a center at and radius of 12.  

Equations for circles are sometimes written in different forms, but we can rearrange them to help find the center and radius of the circle. For example, suppose the equation of a circle is written like this:

We can’t immediately identify the center and radius of the circle. However, if we rewrite the two perfect square trinomials as squared binomials and rewrite the right side in the form , the center and radius will be easier to recognize.

The first 3 terms on the left side, , can be rewritten as . The remaining terms, , can be rewritten as . The right side, 225, can be rewritten as 15². Let’s put it all together.

Now we can see that the center of the circle is  and the circle’s radius measures 15 units.