# The origin of degree scale in angles

#### Aikon

A student showed me a problem, this week, it was: What is the number of regular polygons whose intern angle is an integer number?

I used the equation i=180 (n-2)/n, where i is the internal angle of the polygon and n is the number of the sides of the polygon.

After some trials, i got to the answer. The regular polygon with greater number of sides has n=360 sides and internal angle i = 179 degrees. The answer of the question is that there are 24 polygons with the needed conditions (you need to factorize 360 = 2³ 3² 5 and make the permutations of the exponents, that are 4 (0,1,2,3), and 3 and 2, then you have the total number of divisors of 360).

The interesting thing, I thought today, is that there are 360 sides, this means that when this polygon is limited by a circumference you find that 2 consecutive vertices connected to the center had a 1 degree angle between the lines that connect them. So it appears to me that this is the origin of the degree scale in angles, the polygon with greatest number of sides that have an integer internal angle.

Someone confirms my thesis??

#### Aikon

Thank you,
There are some theorys in:
http://en.wikipedia.org/wiki/Degree_(angle [Broken])

Maybe my theory is a new one??!! : )

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#### uart

Someone confirms my thesis??

I can confirm that yours is a circular argument.

#### HallsofIvy

Homework Helper
A circular argument because you started looking for polygons where the internal angle is an integer number of degrees and determined the largest such had 360 sides and angles of 1 degree.

Had you looked for polygons where the internal angle is an integer number of grads (there are 100 grads in a right angle so 400 grads in a circle), you would have determined that the largest such had 400 sides and angles of 1 grad.

#### Aikon

Yeah i got it...it is a mistake of the type "think a litle more about this...and you will not do".
Well...Living and learning.

#### Makin Bacon

Actually, there are only 22 polygons that satisfy this question.....you cannot have a polygon with 1 side or 2 sides.

#### Aikon

Actually, there are only 22 polygons that satisfy this question.....you cannot have a polygon with 1 side or 2 sides.
Great remark...You are right...Thanks a lot!

Just one thing...I said that i got to the answer...It was a year ago and was a multiple choice question that had answers in the book. Maybe the book was with the wrong answer!

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