Polar coordinates are useful for situations with circular symmetry in the plane. The polar coordinates ($r\text{,}$$\phi$) of a point $P$ are given by the distance $r$ of $P$ from the origin and the angle $\phi$ from the positive $x$-axis to $P\text{,}$ as shown in Figure 1.1.1. Figure 1.1.1. The construction of the polar coordinates ($r\text{,}$$\phi$) at an arbitrary point.
Notation: When we think of the plane as a cross-section of spherical coordinates, we will use the pair ($r\text{,}$$\phi$) for polar coordinates. When we think of the plane as a cross-section of cylindricals coordinates, we will use the pair ($s\text{,}$$\phi$) for polar coordinates. In other references, you may also see the angle called $\theta$ instead of $\phi\text{;}$ we use $\phi$ to agree with our conventions for (cylindrical and) spherical coordinates.
It is important to remember that the angle $\phi$ does not measure distance; it has the wrong dimensions. Angles (in radians) are defined as the ratio of arclength to radius on the circle, so the arclength from the positive $x$-axis to $P$ along the circle shown in the figure is $r\phi\text{.}$