How to Find the Center & Radius of a Circle With a Circle Equation

Use the circle equation to find the center and radius of a circle.

Finding the center and radius of a circle is quite simple, or rather tricky, depending on which circle equation you are working with. If you have a center-radius circle formula, which reads "(x -- h)squared + (y -- k)squared = r squared" where (h,k) is the center and r squared is the radius, then it is simply a matter of reducing the equation to get the answer. The more complex problem arises because a circle equation is often given as "ax squared + by squared+ cx + dy + e = 0." To complete this equation and find the center and radius, you first have to complete the squares of the equation.

Things You'll Need

• Calculator

Instructions

1. Write out the equation with the values in place. For example, "2x squared + 2y squared '8x '12y + 51 = 0." Writing out the equation helps to keep track of the steps involved. First move the loose number, in this case 51, over to the other side of the equation so that it now reads: "2x squared + 2y squared+ 8x + 12y = '51."

2. Group all the "x" and "y" components of the equation together so that it now reads: "2x squared'8x + 2y squared'12y = '51." The terms that are squared in this equation will always be the same amount. In order to get "x" and "y" to their original numerical values (which is the center of the circle) you now divide off the number that is multiplied on the squared "x" and "y" values on every term in the equation. This leaves the equations looking like this: "x squared'4x + y squared'6y = ' 51/2"

3. Rewrite the equation so that you can complete the next phase. It should look like this: "(x squared'4x ) + (y squared'6y ) = ' 51/2." The space is for the squaring term about to be added. Take the coefficient terms (4 and 6) and multiply them by a half to get -2 and -3. Square these numbers and add them to both sides of the equations. You should have: "(x squared'4x+4) + (y squared'6y+9) = ' 51/2+4+9."

4. Convert the left side of the equation to square form so that it reads: "(x'2)squared + (y'3)squared = ' 51/2+4+9." Now reduce the right side to its simplest form. The whole equation should now read: "(x'2)squared + (y'3)squared = square root ¼." Now complete the squares. To find the center of the circle, find (h,k) which is in this case (x,y) and for this example has a value of (2,3) once the squares are completed. The radius is the value on the right of the equation which is now the square root of ¼, which is equal to 1/2.

References

• Purple Math: Completing The Square: Circle Equations