The Polar Coordinate System

Click here to download Maple file for experimenting with polar coordinate system
 
You are probably familar with the cartesian coordinate system, where points are specified by their place along horizontal and vertical axes, represented by X,Y. The polar coordinate system specifies points by the distance from the center (called the radius ("r")), and an angle from the horizontal (usually called theta). Here we see the point 1,45.
It is very easy to draw a circle with polar coordinates: we just say r=1 for theta = 0 to 360. If you wanted only a quater of a circle, you could say r=1 for theta = 0 to 90.

 
It is also easy to draw a spiral with polar coordinates: you can just make r increase as you go around the circle. For example, you can say r=theta. Now every time theta gets bigger, r gets bigger.

Since r is a length, it makes sense to think of theta as how far you have gone around a circle -- a quarter revolution, a half revolution, etc. We can see that this spiral was generated by 3 revolutions.

Rather than call it revolutions, mathematicians use the measure "radians," which is a multiple of the number Pi (3.14159...). One full revolution is 2Pi radians. Since this spiral was made by the equation r=theta, and we had 3 revolutions, the radius at the end of the is 3*2Pi=18.8.