Equation Of A Circle

The equation of a circle tells you which points are a fixed distance from one center point. If a circle has center $(h, k)$ and radius $r$, its standard equation is

because every point $(x, y)$ on the circle is exactly $r$ units from the center. If the center is at the origin, the equation becomes $x^2 + y^2 = r^2$.

When to use this method

Use the standard form whenever you know a circle's center and radius and want its equation, or whenever you want to recognize a circle in coordinate geometry. The expression $x - h$ measures horizontal distance from the center, and $y - k$ measures vertical distance, so squaring those distances and adding them matches the distance formula:

For points on the circle, that squared distance must equal $r^2$. So the equation is a compact way to say, "every point here stays the same distance from the center." Think of the center as an anchor: a circle is the set of all points that stay exactly one radius away from it. That is why the radius matters so much. If you change $r$, the center stays the same but the circle grows or shrinks. There are special cases to keep in mind: if $r > 0$ the equation describes an actual circle, if $r = 0$ it describes exactly one point (the center), and if $r^2 < 0$ there is no real circle, because squared distances cannot be negative.

The procedure, step by step

1. Find the center. Identify the center $(h, k)$ of the circle.
2. Find the radius. Use the given radius $r$, and square it carefully to get $r^2$.
3. Substitute into the form. Write $(x - h)^2 + (y - k)^2 = r^2$ and plug in the values.
4. Check the signs. A center like $(-3, 2)$ becomes $(x + 3)^2 + (y - 2)^2$.
5. Verify one point if needed. A point on the circle should make the equation true and should be exactly $r$ units from the center.

A full example from start to finish

Write the equation of the circle with center $(3, -2)$ and radius $5$.

Start with the standard form:

Substitute $h = 3$, $k = -2$, and $r = 5$:

Simplify:

That is the equation of the circle. Sanity-check it with a point that should lie on the circle. The point $(8, -2)$ is $5$ units to the right of the center, so it should work:

It does, so the equation is consistent with the center and radius.

Where students get stuck, and how to check each step

A few steps cause most errors:

1. Reading the center directly from the signs. In $(x - 3)^2 + (y + 2)^2 = 25$, the center is $(3, -2)$, not $(3, 2)$.
2. Forgetting to square the radius. If the radius is $5$, the right side is $25$, not $5$.
3. Using the diameter as if it were the radius. If the diameter is given, divide by $2$ first.
4. Expecting a real circle when $r^2$ is negative. An equation like $(x - 1)^2 + (y + 4)^2 = -9$ has no real points.

When you look at $(x - h)^2 + (y - k)^2 = r^2$, ask two quick questions: what center do the signs imply, and is the right side really the radius squared? Those two checks catch most errors. The equation of a circle appears in coordinate geometry, analytic geometry, and precalculus, and connects naturally to the distance formula and to completing the square, which is often how you convert a longer equation into standard circle form.

Practice this procedure

Run the steps to write the equation of the circle with center $(-4, 1)$ and radius $3$, then check whether the point $(-1, 1)$ lies on it. To go one step further, start from a longer equation and rewrite it into standard circle form.