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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?

P K
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    There is no mathematical reason. Although according to Wikipedia even the historical reason for it is unclear. – froggie May 08 '12 at 16:55
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    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. ain't a mathematician May 08 '12 at 16:56
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    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 08 '12 at 17:02
  • I like the length of year explanation. Somehow it seems more likely to me. – copper.hat May 08 '12 at 18:12
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    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 08 '12 at 18:14
  • The Ancient Chinese had a system with 365.24 degrees according to Wikipedia – Matthew Finlay May 08 '12 at 23:08
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    yep, there are 400 grads in a circle! This is why you have D/R/G modes on many calculators. – Ronald May 09 '12 at 00:06
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    Another way of measuring angles is the mil: http://en.wikipedia.org/wiki/Angular_mil – JRN May 12 '12 at 07:23
  • @J.M., it’s not really true that all the cools use radians, unless you think an astronomer is automatically uncool. – Lubin Nov 18 '12 at 07:06
  • @Lubin, well, they're weird. ;) (FWIW, I'd almost always use degrees when I'm not talking to someone with scientific sophistication, e.g. carpenters.) – J. M. ain't a mathematician May 14 '13 at 17:47
  • 360° is not possible because according to definition of degree measure, it is $0 < x < 180$ x=real number. That is why we use radian measure! – Freddy Aug 16 '14 at 06:24
  • Does anything else have as many factors? –  Jan 03 '15 at 08:15
  • I once raised this question in one of my math classes (Calculus) and got the (serious) answer "Because $180^°$ is a half circle". Kind of broke my faith in humanity. -- I later told this story, with that reply, to another of my math classes (Precalculus), and one student quipped "hmm, circular reasoning". Faith restored. – Torsten Schoeneberg Dec 19 '22 at 02:47

6 Answers6

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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!

Michael Hardy
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Shaun Ault
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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 09 '12 at 00:44
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    @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. – JRN May 09 '12 at 10:06
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    @JoelReyesNoche Yes of course, but then why didn't we start with the circumference being $\tau$r? Tau manifesto – Zenon May 09 '12 at 15:54
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    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
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    @Zenon: Personally I think $A = \pi r^2$, $C = 2\pi r$ is nicer than $A = \frac{1}{4} \tau r^2$, $C = \tau r$, if only because it avoids the fraction. – Joren May 14 '13 at 11:43
  • Of course that should be 1/2, not 1/4. – Joren May 14 '13 at 11:55
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    It's always seemed to me that turns are the most natural unit of angles. – Doug McClean May 14 '13 at 14:50
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    @DougMcClean Or revolutions, abbreviated "rev", since turn can be ambiguous and mean a right angle turn. – DanielV Aug 16 '14 at 07:10
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    @Zenon because diameter is much easier to measure than radius (you don't have to find the center of a cylindrical tube, for example). And at that time when $\pi$ was introduced, measurement was rather important for geometers. – Ruslan Jan 13 '15 at 09:37
  • @Ruslan, When I took a geometry class, we used a compass, which normally measures radii that is to say much easier than trying to work with diameter. In mathematics, circles are based one radius from a point, the not the diameter. – John Nicholson Mar 17 '15 at 02:16
  • @JohnNicholson that's certainly true when you know the location of center. This is however wrong when you measure properties of e.g. a cylindrical beam or column. And architecture was a major discipline where mathematics was needed when notion of $\pi$ was created. – Ruslan Mar 17 '15 at 09:31
  • @Ruslan, These "measure properties of e.g. a cylindrical beam or column" are born in the hands of a person using the radius to measure and draw these on the architect's paper. It is later when the item is built that your comments hold. But, pure mathematics should work with the fundamentals, not doing so looses the beauty in mathematics. In this stand point, $\pi$ is ugly to $\tau$. To see what I mean, just think how rough a wheel shaped by one $\pi$ radian instead of two. – John Nicholson Mar 17 '15 at 11:19
  • So why not 720 or 3600? https://detechter.com/why-does-a-circle-have-360-degrees/ – user2934433 Oct 16 '23 at 06:18
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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 .

Michael Hardy
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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 08 '12 at 19:51
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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.

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user170010
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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.

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ahala
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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".

J. M. ain't a mathematician
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Nico
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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.

jjrv
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