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I have two related questions:

1- Why do we have 360 degrees in a circle?

2- I have seen in most of the mathematical concepts, angle is expressed in radians not in degrees. Why was radian introduced?

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marked as duplicate by J. M., Amzoti, O.L., Lord_Farin, Henry T. Horton Jun 7 '13 at 17:59

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A good place to start is Wikipedia: en.wikipedia.org/wiki/Degree_(angle), en.wikipedia.org/wiki/Radian. –  Jonas Meyer Jan 31 '13 at 6:48
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We don't need radians. But if we insist on using degrees, many important formulas, particularly in any field that uses calculus, get more complicated, In the same way, we don't need the usual metric system, we could use inches, feet, yards, furlongs, miles. But that is far messier. –  André Nicolas Jan 31 '13 at 6:50

3 Answers 3

Possibly the most important reason for using radians instead of degrees to measure angles is that it's the simplest way for computing the straight length subtended by a circular arc: if we have a circle of radius $r$, then we know that the length of the arc subtended by an angle $\theta$ measured in radians is just $r\times\theta$ - a relation that doesn't hold true if we measure $\theta$ in any other units. This in turn leads to a lot of observations that have important uses in 'higher' mathematics; for instance, the idea that a small piece of circular arc is approximately straight can be formalized as the statement that $\sin\theta\approx \theta$ for angles $\theta$ near $0$ - but only if we measure $\theta$ in radians! A ton of important formulas can then be derived from versions of this formula: for instance, it yields the formula $\dfrac{d}{d\theta}\sin\theta=\cos\theta$, which doesn't hold true if we measure $\theta$ in degrees (it's off by a constant factor), and the list goes on quite a bit. Radians are just, in an important sense, the most natural scaling factor for talking about angles.

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It is presumed that the Babylonians invented the division of the circle into 360 degrees. It's not clear why they used 360. One theory is that back in early Sumerian times they divided the day into 12 "time-miles" - the time required to travel a Babylonian mile. Since a day is one revolution of the sky, that corresponds to dividing a complete revolution into 12 parts. For convenience, the Babylonian mile was subdivided into 30 parts, and that gives 360 subdivisions of the circle. See e.g. Eves, "An Introduction to the History of Mathematics", sec. 2-4.

Hipparchus is said to have introduced the degree into Greece (see http://www-history.mcs.st-andrews.ac.uk/Biographies/Hipparchus.html). The term radian dates back only to about 1869: see http://jeff560.tripod.com/r.html and http://www.jstor.org/stable/3620383

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A circle is defined to have $360$ degrees. That number is convenient because it has many factors and can be divided easily without getting fractions.

Radians are very useful when working with trigonometric functions. Since the input of a trigonometric function can be thought of as a fraction of the circumference of the unit circle, it would seem natural to have the output be in those same units as well.

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