How to graph circles using the center and radius
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=r^2???, where ???(h,k)??? is the center of the circle, and ???r??? is the radius.
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.
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?
We need to find the equation of a circle in the form ???(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.
The center is at ???(-1,-2)??? so ???h=-1??? and ???k=-2???. Now let’s count from the center to a point on the circumference to find the length of the radius. The radius is ???4???, so ???r=4???.
Now let’s plug everything into the standard form of a circle.
???(x-h)^2+(y-k)^2=r^2???
???(x-(-1))^2+(y-(-2))^2=4^2???
???(x+1)^2+(y+2)^2=16???
Example
Graph the circle.
???(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=r^2???
where ???(h,k)??? is the center and ???r??? 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=3^2???
Now we can see that the center is ???(h,k)=(2,-3)??? and the radius is ???r=3???. Let’s graph the circle, starting with the center point.
Since the radius is ???r=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.