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Equation of a Circle

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


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.



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:

Common Core Grade Level/Subject

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