Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1.   Memorize the definition of the polar coordinate system.
2.   Graph points in the polar coordinate system.
3.   Change polar coordinates into rectangular coordinates.
4.   Change rectangular coordinates into polar coordinates.

So far, we have been representing graphs of equations as a collection of points (x, y) in the Rectangular (Cartesian) Coordinate System. For example, in the picture below, we see the point (x, y) being (4, 3).

Now, we will study a second system called the Polar Coordinate System. The initial motivation for the introduction of the Polar Coordinate System was the study of circular and orbital motion. Polar coordinates are now used in navigation and for many phenomena in the physical world.

The Polar Coordinate System

Just like in the rectangular coordinate system we draw a horizontal and a vertical line that intersect at a right angle.The point where the lines intersect is no longer called the Origin, but the Pole.

Instead of rectangular gridlines, we now draw angles theta.gif (860 bytes) with their vertices at the Pole and their initial side at the polar axis. Their terminal sides are called r-axes. Equally spaced circles represent the "grid lines."

A point in the Polar Coordinate System is represented by the polar coordinates . The number r lies on an r-axis which is the terminal side of angle theta.gif (860 bytes). For example, in the picture below, we see the point being (4, 45o).


1. The r-axis is positive along the terminal side of the angle theta.gif (860 bytes) and negative along the EXTENSION of the terminal side through the pole.

2. r and theta.gif (860 bytes) can be positive or negative.

3. theta.gif (860 bytes) can be expressed in radians or degrees.

Strategy for Graphing Points in the Polar Coordinate Systems - see #1 and 2 in the "Examples" document

In a Rectangular Coordinate System, there exists exactly one set of coordinates for each point. On the other hand, in a Polar Coordinate System, a point has infinitely many sets of coordinates.

Formulas for Changing Rectangular Coordinates into Polar Coordinates and Vice Versa

It is often necessary to transform from rectangular to polar form or vice versa. The following relationships are useful in this regard.



and the standard form of the equation of a circle with center at the origin:

They are derived from the following picture:

Strategy for Changing Polar Coordinates into Rectangular Coordinates - see #3 and 4 in the "Examples" document

Strategy for Changing Rectangular Coordinates into Polar Coordinates - see #5 through 8 in the "Examples" document