Center and radius of a circle in standard form

Key takeaway: A circle in standard form gives the center and radius immediately by comparing coefficients and signs.

Match the equation to circle form

The equation is

$(x-4)^2+(y+5)^2=9.$

This already matches the standard circle form

$(x-h)^2+(y-k)^2=r^2.$

Read the center

From $x-4$, we get $h=4$. From $y+5$, we get $k=-5$.

So the center is

$(4,-5).$

Read the radius

Because the right side is 9,

$r^2=9 \quad\Rightarrow\quad r=3.$

So the radius is

$3.$

Key observation

Nothing needs to be expanded here. The equation is already in standard form, so the solution is a direct pattern match. The sign inside each parenthesis is the opposite of the coordinate in the center.

Final answer

• Center: $(4,-5)$
• Radius: $3$

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Pitfalls the pros know 👇 A classic mistake is to swap the signs and write the center as $(-4,5)$. That would come from reading the equation literally instead of translating it into the standard form $(x-h)^2+(y-k)^2=r^2$. Another error is to report the radius as 9 because 9 appears on the right-hand side. The radius is the square root of 9, which is 3.

What if the problem changes? If the equation were

$(x+2)^2+(y-1)^2=100,$

the center would be $(-2,1)$ and the radius would be 10. If the right-hand side changed to 121, the center would stay the same but the radius would become 11. Standard form makes those changes easy to read because the center comes from the parentheses and the radius comes from the square root of the constant.

Tags: circle standard form, center coordinates, radius calculation