Circles

A Circle Is a Distance Rule

A circle is not really about roundness. It is about distance.

A circle is the set of all points that sit the same distance from a fixed center.

That fixed distance is the radius $r$, and the fixed point is the center $(h, k)$.

As the point travels around, the horizontal gap $x - h$ and the vertical gap $y - k$ keep changing, but the radius (the hypotenuse) never does. Their squares always add up to $r^2$.

The Equation

Every point $(x, y)$ on the circle is exactly distance $r$ from the center:

$\sqrt{(x - h)^2 + (y - k)^2} = r$

Square both sides and the roots disappear, leaving the standard equation of a circle:

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

It is just the distance formula with the distance pinned to $r$.

Reading Off Center and Radius

The equation hands you everything if you read the signs carefully.

$(x - 2)^2 + (y + 3)^2 = 25$

• Center: $(2, -3)$
• Radius: $\sqrt{25} = 5$

Watch the signs. $(y + 3)$ means $(y - (-3))$, so the center’s $y$ is $-3$, not $+3$. The equation always subtracts the center coordinates.

Centered at the Origin

When the center is $(0, 0)$, the $h$ and $k$ vanish and the equation shrinks to its simplest form:

$x^2 + y^2 = r^2$

You have already met this one: it is the unit circle when $r = 1$.

Example

Write the equation of the circle with center $(1, 4)$ and radius $2$.

Drop the values straight into the standard form:

$(x - 1)^2 + (y - 4)^2 = 4$

Remember to square the radius: $r = 2$ gives $r^2 = 4$ on the right.