Reading center and radius from standard circle form

Key takeaway: When a circle is already written in standard form, the center and radius can be read off immediately without completing the square.

Use the standard form

The equation

$(x+5)^2+(y+2)^2=324$

matches the standard circle form

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

Read the center

Compare terms carefully:

• $x+5=x-(-5)$, so $h=-5$
• $y+2=y-(-2)$, so $k=-2$

Therefore the center is

$(-5,-2).$

Read the radius

Since

$r^2=324,$

the radius is

$r=\sqrt{324}=18.$

Why this is the fastest method

Because the equation is already in standard form, there is no need to expand or complete the square. The key is to remember that the sign inside the parentheses is opposite the center coordinate.

Check your interpretation

The point $(-5,-2)$ is the center, not $(5,2)$. The radius is positive, so you always take the square root of the constant term and do not keep the square. That gives a clean and direct answer.

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Pitfalls the pros know 👇 Students often misread $(x+5)^2$ as a center x-coordinate of $5$ instead of $-5$. The correct rule is to reverse the sign inside the parentheses. Another pitfall is forgetting that the radius is always nonnegative, so even if the equation suggests a square, the final radius must be the positive square root. Here that means $18$, not $324$ or $-18$.

What if the problem changes? If the equation were

$(x-1)^2+(y-6)^2=49,$

the center would be $(1,6)$ and the radius would be 7. If the right-hand side were not a perfect square, such as $50$, you would still take the square root and leave the radius as $\sqrt{50}=5\sqrt{2}$. The identification process stays the same in every standard-form circle equation.

Tags: standard circle form, radius from equation, circle center coordinates