I have been wondering about the following:
Considering the circumference of a “Perfect Circle” with a Diameter of 1 meter would be something like 3.14 meters, why do we use the number 360 to represent the number of “degrees’ within that circumference?
Would it not make more sense to express the degrees in reference to the relationship to the diameter as related to pi?
That is, let’s just say our “Perfect Circle” has a circumference of 3.14 meters, therefore, what we now consider as due east would change from 90 Degrees to 78.5 Degrees.
Does this make sense to anyone?
I just thought I would ask.
Thanx,
Jack
Hi Jack,
Mathematicians have been measuring angles almost exactly as you suggest since about 1871. The only difference is that we take the reference circle as the circle of radius 1 rather than your choice of the circle with diameter 1. The circumference is then 2 
and hence the 90 degree angle you mention is one-quarter the way around the circle and is hence 2 /4 = /2 units. We call the units radians rather than degrees, the word degree is reserved for the angle measurement based on 360 units.
This is a very astute observation on your part. The radian measure for angles is much more natural than the degree measure, especially when you are using calculus. Degrees come from the Babylonians. You can find a note on this in Chris' answer to a previous question.
I found the 1871 date in Earliest Known Uses of Some of the Words of Mathematics. Look for radians under R.
Penny