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Degrees in a Circle

Date: 09/22/97 at 18:28:12
From: Jane Doe
Subject: Properties of circles

I would like to know why a circle measures 360 degrees. Is there any 
special reason for this calculation, or did the Greeks just kind of 
pick it out? I'm sure there's a rational explanation, but I just can't 
seem to figure it out. I hate accepting things that I don't 
understand, and this is something that really bugs me. Please help!

Date: 09/23/97 at 04:55:18
From: Doctor Pete
Subject: Re: Properties of circles


A circle has 360 degrees, but it also has 400 gradients and 
approximately 6.2831853 radians. It all depends on what *units* you 
measure your angles with.  

Allow me to explain. Say you think 360 is a terrible number, and you 
think that you want a circle to have 100 "somethings" in it. Well, you 
divide up the circle into 100 equal angles, all coming out from the 
center, and then you call one of these angles a "deeg."  Then you've 
just defined a new way to measure a circle. 100 deegs are in a circle.  

This invented unit, the deeg, is much like the degree, except the 
degree is smaller (why?). They are both angles. Just as 1 inch = 2.54 
centimeters, although the centimeter is smaller, the inch and 
centimeter are both units of length. So the ancient Babylonians (not 
the Greeks), decided that a circle should contain 360 degrees.  In one 
degree there are 60 minutes (though they have the same name, one 
minute-angle is not the same as one minute-time). Furthermore, in one 
minute there are 60 seconds (again, one second-angle is not one 
second-time, though they are the same word). 

The British military chose a different way to divide the circle, 
specifically, 400 gradients in one circle. So one gradient is a tad 
bit smaller than a degree. And what's a radian? It's what 
mathematicians use because there's a way to divide the circle into a 
number of parts that happen to make certain computations easy. The way 
they decided this was that they took a circle, say with radius 1 cm.  
Then they took a piece of string, and made marks on it, evenly spaced 
1 cm apart. Then they took the string and wrapped it around the 
circle. They then asked how many little 1 cm pieces of string fit 
around the circle, and they got the answer of about 6.2831853 
pieces. They decided that the angle that a 1 cm piece of string covers 
as it is wrapped about the edge of a circle of radius 1 cm should be 
called one radian. Weird but true.

Now, one might wonder why the Babylonians chose the number 360. The 
reason is that their number system was based on the number 60. To 
compare, we base our number system on 10. For us, 10 is a nice, round 
number and we find it very convenient to count in multiples of 10, 
like millimeter, centimeter, meter, kilometer, etc. But the 
Babylonians liked 60. 

Why this was nice for them, nobody knows, but 60 is a nice number too, 
because 60 = 2 x 2 x 3 x 5 and 360 = 2 x 2 x 2 x 3 x 3 x 5. What's so 
neat about that, you ask?  Well, you will find that 360 is divisible 
by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, and 20. There are few other 
numbers as small as 360 that have so many different factors. This 
makes the degree a very nice unit to divide the circle into an equal 
number of parts. 120 degrees is 1/3 of a circle, 90 degrees is 1/4, 
and so on.

So while a deeg, being 1/100th of a circle, may seem nice and round to 
us ten-fingered folks, it isn't so convenient for dividing a pie into 
thirds.  I mean, who ever heard of 33-and-a-third deegs for a piece of 
pie?  I certainly haven't.

-Doctor Pete,  The Math Forum
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Associated Topics:
High School Conic Sections/Circles
High School Geometry
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Terms/Units of Measurement

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