The origin of degree scale in angles

In summary, the conversation discussed the problem of finding the number of regular polygons with an integer internal angle. The equation i=180 (n-2)/n was used and it was determined that the regular polygon with the greatest number of sides, 360 sides, had an internal angle of 179 degrees. It was also mentioned that this polygon has angles of 1 degree between consecutive vertices connected to the center, which may be the origin of the degree scale in angles. However, it was pointed out that this is a circular argument and considering polygons with integer number of grads may lead to a different result. Lastly, it was noted that there are only 22 polygons that satisfy this question as a polygon cannot have 1 or 2 sides
  • #1
Aikon
21
0
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??
 
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  • #3
Thank you,
There are some theories in:
http://en.wikipedia.org/wiki/Degree_(angle [Broken])

Maybe my theory is a new one??! : )
 
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  • #4
Someone confirms my thesis??

I can confirm that yours is a circular argument.
 
  • #5
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.
 
  • #6
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.
 
  • #7
Actually, there are only 22 polygons that satisfy this question...you cannot have a polygon with 1 side or 2 sides.
 
  • #8
Makin Bacon said:
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!
 

What is the origin of the degree scale in angles?

The degree scale in angles is based on the concept of dividing a circle into 360 equal parts, dating back to ancient civilizations such as the Sumerians and Egyptians.

Why are angles measured in degrees?

The use of 360 degrees to measure angles is likely due to the fact that it is a highly composite number, meaning it has many factors and can easily be divided into smaller parts.

Who first introduced the degree scale in angles?

The Greek mathematician Hipparchus is often credited with introducing the degree scale in angles in the 2nd century BC, although it is possible that other civilizations used a similar concept before this time.

How is the degree scale in angles related to trigonometric functions?

The degree scale is closely related to trigonometric functions such as sine, cosine, and tangent, which are used to calculate the lengths of sides and angles in a right triangle. These functions are based on the ratios of the sides of a right triangle and are measured in degrees.

Can angles be measured in other units besides degrees?

While degrees are the most commonly used unit for measuring angles, other units such as radians and gradians are also used in certain fields of mathematics and engineering. Radians are based on the radius of a circle, while gradians divide a circle into 400 equal parts.

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