Equation of a Circle

Every circle can be written as an equation in the form:

$(x-h)^2+(y-k)^2=r^2$

in which the coordinate $(h,k)$ is the center of the circle and $r$ is the radius of the circle. Any point $(x,y)$ that makes the equation true is on the circle.

When you are given the equation of a circle, you can pull a lot of information from that equation, and even draw the circle based on the information that you pull out.

Example: 

$(x-3)^2+(y-2)^2=16$

Looking at this equation, you know that the center of the circle is $(3,2)$.

Graph that point: 

Coordinate plane with one point (h,k)


Then, pull the radius.  $r^2=16$, so $r=4$.  To graph the circle, plot 4 points that are 4 units from the center (these will help you draw your circle).

Coordinate plane with (h,k) and points on circle

Finally, connect those 4 points to draw the circle: 

Coordinate plane with graphed circle

You can also write the equation of a circle by pulling the center, any point on the circle, and/or, the radius.  

Practice Problems:

  • Equation of a Circle

    Using the following equations, find the center and radius of the circle:

    1. $(x-5)^2+(y-9)^2=81$
    2. $(x-2)^2+(y-4)^2=25$
    3. $(x+4)^2+(y-7)^2=49$
    4. $(x+6)^2+(y+1)^2=16$

    Graph the following circles: 

    1. $(x+3)^2+(y+2)^2=4$
    2. $(x-1)^2+(y-1)^2=9$
    3. $(x+2)^2+(y+1)^2=16$

    Test to see if the following points are on the circle with equation $(x-4)^2+(y-5)^2=81$:

    1. $(13,14)$
    2. $(4,-4)$
    3. $(2,2)$
    4. $(0,5)$

    Write the equations for the following circles:

    1. Graphed circle #1


       
    2. Graphed circle #2



       
    3. Graphed circle #3


       
    4. Graphed circle #3


       
    5. Graphed circle #4

    Answer Key:

Test Prep Practice

  • Equation of a Circle

    1. Which of the following is an equation of a circle in the $xy$ plane with center (3,0) and a radius with endpoint (6,4)?
      1. $(x-3)^2+y^2=36$
      2. $(x-3)^2+y^2=25$
      3. $(x-3)^2-y^2=16$
      4. $(x-3)^2-y^2=25$
    2. Which of the following is an equation of a circle in the $xy$ plane with center (0,-2) and a radius with an endpoint on the origin?
      1. $x^2+(y-2)^2=2$
      2. $x^2+(y-2)^2=4$
      3. $x^2+(y+2)^2=2$
      4. $x^2+(y+2)^2=4$
    3. $x^2+y^2-2x+6y=6$ is the equation of a circle in the $xy$ plane. What is the radius of the circle?
      1. 2
      2. 3
      3. 4
      4. 6
    4. $x^2+y^2+12x-10y=-12$ is the equation of a circle in the $xy$ plane. What is the radius of the circle?
      1. 3
      2. 5
      3. 7
      4. 9
    5. $x^2+y^2-8x+2y=-4$ is the equation of a circle in the $xy$ plane. What are the coordinates of the center of the circle?
      1. (-8,2)
      2. (8,-2)
      3. (4,-1)
      4. (-4,1)
    6. $x^2+y^2-3x-5y=-3$ is the equation of a circle in the $xy$ plane. What are the coordinates of the center of the circle?
      1. (1.5, 2.5)
      2. (6,10)
      3. (9,25)
      4. (6,-10)

    Answer Key:

Skill:

Common Core Grade Level/Subject

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