How to graph circles using the center and radius

 
 
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Standard form of the equation of a circle

In this lesson we’ll look at how the equation of a circle in standard form relates to its graph.

Remember that the equation of a circle in standard form is (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2, where (h,k)(h,k) is the center of the circle, and rr is the radius.

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As you can see in the image, the center of a circle is a point and the radius of a circle is the distance from the center of the circle to a point on its circumference.

 
center of the circle (h,k) and radius r
 

This means that if you have a graph of a circle, you can write its equation in standard form.

 
 

How to find the equation of a circle and sketch its graph

 
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Finding the equation from the graph of a circle

Example

What is the equation of the circle shown in the graph?

find the equation of the circle in the graph


We need to find the equation of a circle in the form (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2, which means we need to find the center point and the length of the radius.

Let’s find the center first.

circle sketched in the xy-plane

The center is at (−1,−2)(-1,-2) so h=−1h=-1 and k=−2k=-2. Now let’s count from the center to a point on the circumference to find the length of the radius. The radius is 44, so r=4r=4.

Now let’s plug everything into the standard form of a circle.

(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2

(x−(−1))2+(y−(−2))2=42(x-(-1))^2+(y-(-2))^2=4^2

(x+1)2+(y+2)2=16(x+1)^2+(y+2)^2=16

Graphing circles for Algebra 2.jpg

This means that if you have a graph of a circle, you can write its equation in standard form.

Example

Graph the circle.

(x−2)2+(y+3)2=9(x-2)^2+(y+3)^2=9

In order to graph a circle, we need to know its center and radius. In standard form, the equation of a circle is

(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2

where (h,k)(h,k) is the center and rr is the radius. Let’s go ahead and write out the equation as

(x−2)2+(y+3)2=9(x-2)^2+(y+3)^2=9

(x−2)2+(y+3)2=32(x-2)^2+(y+3)^2=3^2

Now we can see that the center is (h,k)=(2,−3)(h,k)=(2,-3) and the radius is r=3r=3. Let’s graph the circle, starting with the center point.

first graph the center point of the circle

Since the radius is r=3r=3, we’ll count three units in all directions from the center point, or we can use a compass to draw a more perfect circle.

sketch the radius from the center of the circle
 
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