Standard Form

The standard equation for a circle with center ${\displaystyle (h,k)}$ and radius ${\displaystyle r}$ is

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

In the simplest case of a circle whose center is at the origin, the equation is simply a restatement of the Pythagorean Theorem:

${\displaystyle x^{2}+y^{2}=r^{2}}$

General form

The general form of a circle equation is

${\displaystyle x^{2}+y^{2}+2gx+2fy+c=0}$, where

<-g,-f> is the center of the circle.

Polar Coordinates

In the case of a circle centered at the origin, the polar equation of a circle is very simple because polar coordinates are essentially based on circles. For a circle with radius ${\displaystyle a}$,

${\displaystyle r=a}$.

In the more complicated case of a circle with an arbitrary location, the equation is

${\displaystyle r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}}$,
where ${\displaystyle r_{0}}$ is the distance from the circle's center to the origin and ${\displaystyle \varphi }$ is the angle pointing to the circle.

There are many cases that allow the equation to be simplified. If a point on the circle is touching the origin, its polar equation may consist of a single trig function.

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Parametric Equations

When the circle's equation is parametrized with respect to ${\displaystyle t}$, the equation becomes

${\displaystyle x=h+r\cos t}$,
${\displaystyle y=k+r\sin t}$.