Polar coordinates are a form of expressing position on a two-dimensional plane. Cartesian coordinates, also called rectangular coordinates, utilize a distance in each of two dimensions to locate a point, but polar coordinates make use of an angle and a distance. The distance is sometimes referred to as the radius.

Rectangular coordinates typically are denoted *(x,y)*, where *x* and *y* are distances along those respective axes. In a similar manner, polar coordinates are expressed as *(r,θ)*. The letter *r* is the distance from the origin at the angle represented by the Greek letter theta, *θ*, where *r* can be a positive or negative number. If a negative distance is used, the magnitude of the distance does not change, but the direction is taken opposite the angle *θ* on the other side of the origin. A point in a polar coordinate system can be referred to as representing a vector, with a magnitude of *r*, a direction of *θ* and a sense of direction, which is the sign of *r*.

Translation between rectangular and polar coordinates can be accomplished through the use of trigonometric formulas. For conversion from rectangular to polar, the following formulas can be applied: *θ* = tan^{-1}*(y/x)* and *r* = √(*x ^{2}* +

*y*). For changes from polar to rectangular, these equations can be employed:

^{2}*x*=

*r*cos

*θ*and

*y*=

*r*sin

*θ*.

Polar coordinates tend to be used for any situation in which rectangular coordinates would prove difficult or awkward to utilize, and vice versa. Any application involving circular geometry or radial movement is ideally suited to polar coordinates, because these geometries can be described with relatively simple equations in a polar coordinate system; their graphs are more curvilinear or circular in appearance compared to those on rectangular coordinate systems. As a result, polar coordinates have use representing models of real-world phenomena that have similarly rounded shapes.

The applications of polar coordinates are quite varied. Polar coordinate graphs have been used to model the sound fields produced by varying speaker locations or the areas where different types of microphones can best pick up sound. Polar coordinates are of great importance modeling orbital motions in astronomy and space travel. They are also the graphical basis for the famous Euler Formula, which is regularly applied in mathematics for representation and manipulation of complex numbers.

Like their rectangular counterparts, polar coordinates need not be limited to only two dimensions. To express values in three dimensions, a second angle represented by the Greek letter phi, *φ*, can be added to the coordinate system. Any point can be thus located from the origin by a fixed distance and two angles, and it can be assigned the coordinates *(r,θ,φ)*. When this type of nomenclature is used for tracking and locating points in three dimensional space, the coordinate system is designated as a spherical coordinate system. This type of geometry is sometimes referred to as using polar spherical coordinates.

Spherical coordinates actually have a well-known application — they are used in mapping the Earth. The angle *θ* is typically the latitude and is limited to between minus-90 degrees and 90 degrees, whereas the angle *φ* is longitude and is held to between minus-180 and 180 degrees. In this application, *r* can sometimes be ignored, but it is more often employed for the expression of elevation above mean sea level.