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I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number?

What is the significance of 90° for the perpendicular or 360° for a circle?

I didn't ever think about this during my school time.

Can someone please explain it mathematically? Is it due to some historical reason?

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There is no mathematical reason. Although according to Wikipedia even the historical reason for it is unclear. –  froggie May 8 '12 at 16:55
It's a Babylonian convention that we divide a circle into $360^\circ$. For a while, there was the grad measure where a right angle is 100 grads, but it seems to have never caught on. Of course, all the cool kids use radians nowadays. –  J. M. May 8 '12 at 16:56
There has been speculation that it is because $360$ is a "nice" number close to the length of the year in days. Angles (or with greater historical accuracy, arcs of circles) had their primary application in astronomy/astrology. –  André Nicolas May 8 '12 at 17:02
It's worth looking at other cases where we divide a single unit into multiple units, particularly the non-metric units, like 12 inches in a foot, 24 hours in a day, etc. These numbers are, in a sense, arbitrary ways of dividing up a unit, and we often see lots of multiples of small primes in these numbers, so we can frequently get fractions of a unit as integers of another - $1/6$ of a day is $4$ hours, for example. –  Thomas Andrews May 8 '12 at 18:14
yep, there are 400 grads in a circle! This is why you have D/R/G modes on many calculators. –  Ronald May 9 '12 at 0:06

4 Answers 4

up vote 36 down vote accepted

360 is an incredibly abundant number, which means that there are many factors. So it makes it easy to divide the circle into $2, 3, 4, 5, 6, 8, 9, 10, 12,\ldots$ parts. By contrast, 400 gradians cannot even be divided into 3 equal whole-number parts. While this may not necessarily be why 360 was chosen in the first place, it could be one of the reasons we've stuck with the convention.

By the way, when working in radians, we just "live with" the fact that most common angles are fractions involving $\pi$. There's a small group of people who prefer to use a constant called $\tau$, which is just $2\pi$. Then angles seem naturally to be divisions of the circle: The angle that divides a circle into $n$ equal parts is $\tau/n$ (radians).

Hope this helps!

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+1 for \tau! (And as you said, the real question should be why did we use 2\pi in radians? ) –  Zenon May 9 '12 at 0:44
@Zenon, the circumference of a circle is $2\pi r$. If you like the formula for the arclength $s=r\theta$, then you would choose $\theta=2\pi$ for one revolution. –  Joel Reyes Noche May 9 '12 at 10:06
@JoelReyesNoche Yes of course, but then why didn't we start with the circumference being $\tau$r? Tau manifesto –  Zenon May 9 '12 at 15:54
Possibly, besides the fact that 360 is divisible by 2,3,4,5,6, it may have also been chosen historically because it is close to the number of days in an year, and therefore a "meaningful" number when dealing with circles (for Aristotle, any motion "in the heavens" was supposed to be circular) –  Dalker May 20 '12 at 17:58
It's always seemed to me that turns are the most natural unit of angles. –  Doug McClean May 14 '13 at 14:50

I have heard that the ancient Babylonians used a base-$60$ numeral system with sub-base $10$.

Certainly such a system was used by Ptolemy in the second century AD. See Gerald Toomer's translation of Ptolemy's Almagest. In particular Ptolemy divided the circle into $360$ degrees. See http://en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords, http://en.wikipedia.org/wiki/Almagest, and http://hypertextbook.com/eworld/chords.shtml .

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My guess is that the Babylonian system is the main reason for that. It is also worth noting that the smallest regular polygon has all its angles equal to 60 degrees; historically the choice might have been made first for the equilateral triangle... –  N. S. May 8 '12 at 19:51

360 degrees is not the only choice. When using grads (also called gons) as a unit of angle, the full circle is 400 degrees and the right angle is 100 degrees. Grads are used in surveying and for example a theodolite, a surveying instrument, often has its measuring scale labeled in grads. It seems the unit was introduced along the metric system in an attempt to replace historical units, but only caught on in some fields.

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In military (well, I don't know if this is true for all countries), a full turn of a circle is 6400' (I don't know the translation of "milésimos", thousandth?).

This has the advantages already mentioned (divisible by 2, 3, 4, ...) and they can easily use it to measure things. 1 milésimo is about 1 meter at 1 km. So, if they know that a target is at about 1 km and it has 3 milésimos, the target is about 3 meters wide. They use it the other way round too. For example, a tank is about 7-8 meters by 2-3 meters. If they see a tank with their binocular and measure about 7 milésimos, they know that the tank is about 1 km away.

They have several ways to measure milésimos, from graduated binocular to "rules of thumb".

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You mean the mil? A very convenient unit, especially for sniping purposes... –  J. M. May 14 '13 at 17:43
Yes, you can see a picture there using one's hand to measure. –  Nico May 14 '13 at 17:46

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