Why is a full turn of the circle 360°? Why not any other number? 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?
 A: It is from ancient astronomy. A day in earth is a natural unit of time, and one year is also a natural time unit. one year is 360 days in ancient calendar,  People related to circle to year, therefore a circle is divided to 360 degrees. There is not much free choices here if we establish this relationship. 
The accepted answer can explain why people choose 360 over 365, but cannot explain why a circle is not 720 or 3600. 
A: 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!
A: 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".
A: 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 .
A: Ancient civilizations had used a system of counting numbers with fingers on their palms.   Later they found it is easier to count higher numbers with their fingers only but with the folds of their fingers on their palm to 12 excluding the thumb.   Further by folding each finger of the other hand they counted up to 60, that is 12 multiplied by 5.   Old sailors from Greece and southern Europe used to count with their fingers only in a similar way.   Still many people around the world practice this.    British count and measuring system is based on 12 and 60.
A solar day consists of 2 distinct parts; the day - time in light, and the night - time in dark, due to rotation of Earth on its axis and falling sunlight.    A day time is divided in to 12 parts as one can count it with his finger folds of a hand represented for day.   A night time is similarly divided by his finger folds of the other hand represented for night.    Thus one get a sidereal day consist of 24 hours.
An hour is divided by 60 to count a smaller portion of an hour and further a minute is divided by 60 to get a second.
Ancient people knew that climatic conditions were changing and repeating in about 360 days, by observing natural events like migration of birds, mating season of animals, flowering of trees etc.     They divided it in to 4 equal portions by naming seasons.    When they started cultivation and harvesting they needed a calendar consisting of 4 seasons contained in 360 days year.
By observing the stars position they knew it came back every 360 days.    Around 4000 years ago the observers and mathematicians of that time started thinking of the Earth being a spherical mass which spins its own and revolves around the sun, they related the number 360 to a circle and and started to count a smaller portion of a circle to be one  in 360 parts (called one degree).     Thus the degree of a circle or the measurement of angles are in a way  related to our fingers and its folds of our palm. 
A: 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.
