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### Why Is a Circle 360 Degrees?

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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

Chris Perkey
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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?

- Doctor Pete, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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Associated Topics:
Elementary Circles
Elementary Definitions
Elementary Math History/Biography
Elementary Terms & Units of Measurement
High School Conic Sections/Circles
High School Definitions
High School History/Biography
Middle School Conic Sections/Circles
Middle School Definitions
Middle School History/Biography
Middle School Pi
Middle School Terms/Units of Measurement

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