How to Find the Center and the Radius of Circles

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.

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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:

Identify the center and radius.
\(8x+x^2-2y=8- y^2 \)

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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