Why Is a Circle 360 Degrees?

Date: 07/01/98 at 02:31:44
From: Chris Perkey
Subject: Why is a circle defined as 360 degrees?

That's my full question. It came up in class today, and the best 
answer that I had was that maybe the Babylonians did it to coincide 
with their base 60 system. This sort of seems incomplete. Is there a 
better, more complete, or correct answer. Or does anyone know? Thanks 
in advance for your help.  

Chris Perkey

Date: 07/01/98 at 02:57:33
From: Doctor Pete
Subject: Re: Why is a circle defined as 360 degrees?

Hi,

As a matter of wording, it is not so much that the circle is defined 
to be  360 degrees, but the other way around - the degree is a unit 
of angular measure which is 1/360th of a full "rotation." For 
instance, the grad is defined to be a unit of angle in which there are 
400 in a circle. I believe this unit derives from British military 
use. But most important of all is the radian.  There are precisely 2Pi 
radians in a circle, where Pi is the ratio of the circumference to its 
diameter, which is approximately:

   3.14159265358979323846264338327950 ...  

(Sidenote: the importance of Pi lies in the fact that this value is 
constant, regardless of the radius of the circle.)

Anyway, the idea really is that the degree (or grad or radian) is a 
unit that measures angle, much the way the meter measures length, or 
the gram measures mass, or the second measures time. But whereas these 
basic units (m, g, s), are defined in terms of things like atomic 
vibrations, the speed of light, or a piece of metal sitting in vacuum 
in Paris, the measure of angle is a mathematical abstraction. One 
could choose any sort of unit, for instance you could call a "perkey" 
an angle measurement in which there are 51235071 in a circle. Of 
course, this is a bit reminiscent of the days when the "foot" was a 
unit of length corresponding to the size of a king's foot.

So in particular, you are quite correct in that the degree is derived 
from the Babylonian base 60 numerical system. Hours and minutes are 
similarly divided into 60's (of course, there are minutes of time and 
minutes of angle - there are 60 minutes in a degree, and, similarly, 
there are seconds of time and seconds of degree - there are 60 seconds 
in a minute, 3600 in a degree).

And finally, you may wonder why the radian is defined as 2Pi in a 
circle - this number is not even an integer. The reason is that the 
radian satisfies certain nice properties, so that trigonometry and 
calculus are made easier when you let angles be measured in radians. 
The underlying idea is that the circumference of a circle of radius 1 
is 2Pi. So in a circle of radius 1, one radian subtends an arc length 
of exactly 1. This makes measuring arc length equivalent to measuring 
angle. Similarly, in a circle of radius 1, Pi radians (which, by the 
way, equals 180 degrees), subtends an arc length of Pi. And so on and 
so forth. Now, what arc length is subtended by an angle of 2 radians 
in a circle of radius 3?

I hope that's answered your question in more detail!

- Doctor Pete, The Math Forum
Check out our web site! http://mathforum.org/dr.math/