Let’s say that you are an expert in all things related to graphing. Therefore, anytime someone gives you two numbers, you can right away figure out which number corresponds to the *x*-axis, and which to the *y*-axis, and find exactly where to plot the point. If someone gives you three numbers, you can even plot that point on three axes: *x*-axis, *y*-axis, and *z*-axis, thereby figuring out the position of that point in three dimensions. But, what if all of a sudden somebody gives you a set of points that is (3, 60^{o}) and asks you to graph that point? What does this even mean? Why does one of these values look like a normal number and the other one look like an angle? After some initial confusion, you may be able to figure out the truth: this person is talking about polar coordinates, and plotting a point on a graph based on that system.

What are polar coordinates, however, and how are they different from our “normal” coordinates?

### Normal Coordinates

First of all, let’s briefly review what “normal” coordinates are. These are technically referred to as “Cartesian coordinates,” named after the mathematician Rene Descartes, who developed this system. In Cartesian coordinates, each number corresponds to a distance from a particular axis to the point in question. So, for example, the coordinates of (3,5) mean that to get to that point, you need to travel 3 units on the *x*-axis and 5 units on the *y*-axis. The coordinates of (0,3,5) mean that you travel 0 units on the *x*-axis, 3 units on the *y*-axis, and 5 units on the *z*-axis. This system feels very “normal” for most of us, because much of what we have learned in mathematics (i.e., nearly everything) is based on this kind of graphing and plotting system.

### Polar Coordinates

Polar coordinates, however, work very differently. More specifically, each point can be described using two values: the first number describes the distance from the center of the graph (or a different reference point) to the point in question, and the second number describes an angle (measured relative to a “reference” direction). Instead of having an *x*-axis and *y*-axis like in Cartesian coordinates, polar coordinates have a “radial coordinate” (describes the distance) and an angular coordinate (which describes the angle).

Why would anyone use such a bizarre system, you may be wondering? Well, the whole system of polar coordinates would probably seem less “bizarre” if you had learned about this system and used polar coordinates at an early stage in your education. But also, the entire polar coordination system is uniquely suited for describing things that exist or move in spirals, circles, spheres, or other similar trajectories. What are some examples of these kinds of things? Our solar system, for one, which can be described most readily using the distance from a central point (i.e., the sun) and angles that relative to a reference direction, or position. People who navigate ships also use polar coordinates, as did Muslims in the 8^{th} and 9^{th} Centuries who wanted to calculate their positions relative to a religious landmark, namely, the city of Mecca.

### How to Use Them

So now that we understand WHY people may want to use polar coordinates, let’s discuss HOW to use them. The first number is the distance between your point and a central reference point, which is most cases is the origin. This number essentially provides a “circle” where your point could be, with the center of the circle at the origin of the graph. Every point on that circle meets the requirement of being a certain distance from the origin! This is why you need a second number, to pinpoint where on the circle you are meant to be located. This second number is an angle, usually measured between the positive *x*-axis and the line that connects your point to the origin. Once you have both the distance from the origin, and the angle measure, you can figure out where your point is.

### Graphing Geometric Objects with Polar Coordinates

But let’s say you want to graph something more than a single point, like maybe a geometric object (a circle, for example, or a parabola), using polar coordinates. How do you do that? Well, for that you need you need polar equations! What is a polar equation? It is an equation that uses *r* and “theta,” where *r* measures the distance from the reference point and “theta” measures the angle from a reference direction. It’s kind of like a “regular” equation that uses *x* and *y* as variables, but then lets us plot the data on Cartesian axes. We use polar equations to describe more complex functions, and then plot the polar equations directly on polar coordinates. Pretty cool, right?

One final note: we can also convert between Cartesian coordinates and polar coordinates. To do so, we consider that any point plotted with polar coordinates can become part of a right triangle, where the hypotenuse of the triangle is the radius, and the legs of the triangle are the horizontal and vertical distances traveled. Since we already know “theta,” or the angle between our hypotenuse and the reference direction, we have enough information to figure out both the *x *and *y* values (with a little trigonometry and a little Pythagorean theorem).

How’s that for some versatile graphing abilities? I think it’s pretty awesome!

Author: Mindy Levine, PhD