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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.


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