Circle: The equation $r \= a$ creates a circle of radius a centered at the pole. The equation $r\=a\cos \left(\mathrm{\theta }\right)$ creates a circle of diameter a centered at the point $\left(r\,\mathrm{\θ}\right) \= \left(\frac{a}{2}\,0\right)$. The equation $r\=a\sin \left(\mathrm{\theta }\right)$creates a circle of diameter a centered at the point $\left(r\,\mathrm{\theta }\right) \= \left(\frac{a}{2}\,\frac{\mathrm{\π}}{2}\right)$.

Cardioid: The equations $r \= a \+ a\cos \left(\mathrm{\theta }\right)$ and $r \= a \+ a\sin \left(\mathrm{\θ}\right)$ create horizontal and vertical heart-like shapes called cardioids.

Archimedean Spiral: any equation of the form $r\=a{\mathrm{\θ}}^{\frac{1}{n}}$ creates a spiral, with the constant n determining how tightly the spiral winds around the pole. Special cases of Archimedean spirals include: Archimedes' Spiral when $n \= 1$, Fermat's Spiral when $n\=2$, a hyperbolic spiral when $n \= -1$, and a lituus when $n \= -2$.

Polar Rose: Equations of the form $r\=a\cos \left(k\mathrm{\θ}\right)$create curves which look like petaled flowers, where a represents the length of each petal and k determines the number of petals. If k is an odd integer, the rose will have k petals; if k is an even integer, the rose will have $2k$ petals; and if k is rational, but not an integer, a rose-like shape may form with overlapping petals.

Ellipse: Equations of the form $r\&equals;\frac{l}{1\&plus;e\cos \left(\mathrm{\theta }\right)}$, where $-1 < e < 1$ is the eccentricity of the curve (a measure of how much a conic section deviates from being circular) and l is the semi-latus rectum (half the chord parallel to the directrix passing through a focus), create ellipses for which one focus is the pole and the other lies somewhere along the line $\mathrm{\&theta;} \&equals; 0$. The special case where $e \&equals; 0$ creates a circle of radius l.