# How to Find the Center and the Radius of Circles

To find the center and radius of a circle from its equation, you need to find the equation of the circle in is the centerradius form.

## Rules for Finding the Center and the Radius of Circles

To find the center and the radius of a circle using the equation of the circle:

• Write the equation of the circle in standard form: $$(x- h)^2+( y-k)^2= r^2$$,
• The center of the circle is at (h,k), and its radius is r.

## Examples

### Find the Center and the Radius of Circles – Example 1:

Identify the center and the radius of each circle.

$$x^2+ y^2-4x+3=0$$

Solution:

$$(x- h)^2+( y-k)^2= r^2$$ is the circle equation with a radius r, centered at (h,k).
Rewrite $$x^2+ y^2-4x+3=0$$ in the standard form:
$$x^2+ y^2-4x+3=0→(x-2)^2+y^2=1^2$$
Then, the center is at: (2,0) and $$r=1$$

### Find the Center and the Radius of Circles – Example 2:

Identify the center and the radius of each circle.

$$8x+x^2+10y=8- y^2$$

Solution:

Rewrite the equation in standard form:
$$8x+x^2+10y=8- y^2→(x-(-4))^2+(y-(-5))^2=7^2$$
Then, the center is at $$(-4,-5)$$ and the radius is 7.

### Find the Center and the Radius of Circles – Example 3:

$$8x+x^2-2y=8- y^2$$

Solution:

$$(x- h)^2+( y-k)^2= r^2$$ is the circle equation with a radius r, centered at (h,k).
Rewrite $$8x+x^2-2y=8- y^2$$ in the standard form:
$$(x-(-4))^2+(y-1)^2=5^2$$
Then, the center is at $$(-4,1)$$ and $$r=5$$

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