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The Equation of a Circle

Every circle in the coordinate plane has a tidy equation: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center and rr is the radius. It comes straight from the distance formula — every point on the circle is exactly rr away from the center.

The one thing that trips people up is the signs. The center coordinates appear inside the parentheses with the opposite sign, and the radius is squared. Get those two habits down and reading a circle equation is automatic.

Reading the center and radius

In (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, the center is (h,k)(h, k) — the numbers that make each parenthesis zero. Because the form subtracts, a center of (3,−2)(3, -2) shows up as (x−3)2+(y+2)2(x - 3)^2 + (y + 2)^2: the −2-2 flips to +2+2.

The right side is r2r^2, not rr. If the equation ends in 2525, the radius is 25=5\sqrt{25} = 5. Forgetting to take the square root is the most common mistake.

The figure shows what the equation describes: a circle centered at (h,k)(h, k) whose every point sits a distance rr from that center.

rr
(h, k)(h,\,k)

Writing the equation from a center and radius

Go the other direction by substituting. Center (h,k)(h, k) drops into the parentheses with flipped signs, and the radius gets squared on the right.

For center (3,−2)(3, -2) and radius 55: (x−3)2+(y+2)2=25(x - 3)^2 + (y + 2)^2 = 25. Build it piece by piece and the signs take care of themselves.

Worked examples

Example 1: center and radius from an equation

Find the center and radius of (x−4)2+(y+1)2=9(x - 4)^2 + (y + 1)^2 = 9.

Match to standard form(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
Read the center, flipping signs(h,k)=(4,−1)(h, k) = (4, -1)
Take the square root of the right sider=9=3r = \sqrt{9} = 3

Answer: Center (4,−1)(4, -1), radius 33

Example 2: equation from center and radius

Write the equation of the circle with center (−2,5)(-2, 5) and radius 44.

Substitute the center with flipped signs(x+2)2+(y−5)2=r2(x + 2)^2 + (y - 5)^2 = r^2
Square the radiusr2=16r^2 = 16
Write the equation(x+2)2+(y−5)2=16(x + 2)^2 + (y - 5)^2 = 16

Answer: (x+2)2+(y−5)2=16(x + 2)^2 + (y - 5)^2 = 16

Try one yourself

Common questions

Why do the center's signs flip?

The form subtracts the center coordinates. To make (x−h)(x - h) equal zero at the center's x-value, hh must be that value — so a center at x=−2x = -2 appears as (x−(−2))=(x+2)(x - (-2)) = (x + 2).

What if the right side is not a perfect square?

The radius is still the square root — just leave it in radical form. If the equation ends in 2020, the radius is 20=25\sqrt{20} = 2\sqrt{5}.

How is this related to the distance formula?

It is the distance formula. Setting the distance from (x,y)(x, y) to the center (h,k)(h, k) equal to rr and squaring both sides gives (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 directly.

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