Equation of a circle

The standard form of an equation of a circle is ( x - h )2 + ( y - k )2 = r2. The radius is r, the center of the circle is (h , k), and (x , y) is any point on the circle.

For example, suppose ( x - 2 )2 + ( y - 3 )2 = 42 is an equation of a circle. The center of this circle is located at ( 2 , 3 ) on the coordinate system and the radius is 4.

How to derive the standard form of an equation of a circle.

Start with the circle you see below. Then put the circle on the coordinate system.

Finally, label the circle to show the center and a point on the circle. Recall that (h, k) is the center of the circle and (x, y) is a point on the circle. The distance between (h, k) and (x, y) is the length of the radius.


To find the equation, we just need to use the distance formula which can be used to find the distance between two points.

  __________________
Distance formula = √  (x1 - x2)2 + (y1 - y2)2  


  __________________
d = √  (x1 - x2)2 + (y1 - y2)2  


Notice that the distance between (h,k) and (x,y) is r. Thus d = r

  __________________
r = √  (x1 - x2)2 + (y1 - y2)2  


Let (x1, y1) = (x,y) and (x2, y2) = (h,k)

  __________________
r = √  (x - h)2 + (y - k)2  


Square each side of the equation

r2 = ( x - h )2 + ( y - k )2

Writing the equation of a circle

Write the standard equation of the circle with center (4, -1) and a radius of 6.

The standard form is (x - h)2 + (y - k)2 = r2

Substitute (4, -1) for (h, k) and 6 for r.

(x - 4)2 + [(y - (-1)]2 = 62

Simplify the equation

(x - 4)2 + (y + 1)2 = 36

Recent Articles

  1. Find the Multiplicity of a Zero

    Oct 20, 21 04:45 AM

    Learn how to find the multiplicity of a zero with this easy to follow lesson

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.