Equation of Circle

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Theorem

The equation of a circle with radius $R$ and center $\tuple {a, b}$ can be expressed in the following forms:

Cartesian Coordinates

$\paren {x - a}^2 + \paren {y - b}^2 = R^2$


Parametric Equation

$x = a + R \cos t, \ y = b + R \sin t$


Polar Coordinates

In polar coordinates, it does not make sense to refer to a point by $x$ and $y$ coordinates.

Instead, the center of a circle is commonly denoted $\polar {r_0, \varphi}$, where $r_0$ is the distance from the origin and $\varphi$ is the angle from the polar axis in the counterclockwise direction.

The equation for a circle with radius $R$ of this type is

$r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$