Zero Degree and 360 Degree Angles

Date: 10/22/2003 at 21:08:51
From: Katie
Subject: Is there such a thing as a zero degree angle?

I am curious if there is such a thing as a zero degree angle, and if 
so, what does it look like?  I am also wondering if a zero degree 
angle equals a 360 degree angle?  I understand that a 360 degree angle 
is essentially a circle, or the amount of "turn" that equals a circle.  
I am pondering how you would draw a 360 degree angle.  Would it be 
drawn like ___________ (or just a straight line)?  

Date: 10/22/2003 at 23:14:52
From: Doctor Peterson
Subject: Re: Is there such a thing as a zero degree angle?

Hi, Katie.

These are some good questions.  We've dealt with some similar issues here:

  Angles as Turns
    http://mathforum.org/library/drmath/view/62997.html 

  Angles Greater than 360 Degrees
    http://mathforum.org/library/drmath/view/55067.html 

Angles can be thought of in several different ways.  One is as a
figure consisting of two rays (not lines) starting at the same point:

                   /
                  /
                 /
                /
               o--------->

In that sense, a 0 degree angle would be a "degenerate angle", meaning
one that no longer quite fits the definition, since it is only one
ray, not two:

               o--------->

And angles, thought of this way, can only have a measure less than 180
degrees, since we always measure the short way around.

But another way to think of an angle is as the "space" between two
rays, so that our first figure includes two angles, one on the
"inside" and the other on the "outside"; one less than 180 degrees,
and one greater.  In that sense, our second figure shows both a 0
degree angle and a 360 degree angle, and they are different parts of
the figure.

Thirdly, an angle can be thought of as a rotation, as if we started 
at one of the rays and turned it to the other.  Then our figure
represents the result of many possible angles, starting at either ray 
and going clockwise (which we give a negative measure) or 
counterclockwise (which we give a positive measure) until we reach the 
other ray.  That distance we rotate may be just part of a circle or 
more than one time around (60, -60, 300, -300, 420 degrees, and so 
on).  In this view, our single ray can be seen as 0, 360, -360, 720 
degrees, and any other multiple of 360 degrees.

So, does a 0 degree angle equal a 360 degree angle? Only in the first 
sense of the three.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/