Unit 40: Center Radius Form of a Circle

If you don’t know it already, this is another formula you will have to memorize.  It describes every point on a circle, as well as the center and radius of the circle.

$$boldsymbol(boldsymbol xboldsymbol-boldsymbol hboldsymbol)^{mathbf2}boldsymbol+boldsymbol(boldsymbol yboldsymbol-boldsymbol kboldsymbol)^{mathbf2}boldsymbol=boldsymbol r^{mathbf2}$$

(h,k) describes the x and y coordinates of the center of the circle, and r is simply the radius.  x and y in the equation represent any point going around the circle.

If you were given the equation of a circle that looked like this:


You would know the center of the circle is at (0,0), and it has a radius of 12.  That equation is the same as:


Our h and k values are thus 0, and the radius is (sqrt{144}) or 12.


  • Know the formula and understand all components of it.

Another example that is slightly more difficult:


Here, our center would be at (2,-3) and the radius would be (sqrt{50}).  Note the y value is negative because it is (y+3) in the equation, where our standard form says (y-k).

If you are given something that looks close to, but not quite standard form, you need to manipulate it until you get it looking just like standard form.


We would first add 20 to both sides, then divide by 2:


Now you can determine the center and radius for this circle.

What if you are given a graph and asked what the equation of the circle is?

Easy!  We just find the center of the circle is (-2, 2) and count out that the radius is 3.  Since we have our formula memorized, we just plug in those three values to get


In the next unit, we’ll cover one more type of circle problem (the hardest kind!).