Mathematics Magazine
Home Articles Math Book Applications Contact Us
Bookmark and Share Advertise Here. E-mail us your request for an advertising quote!

Radians Degrees Gradians

by Liliana Usvat


The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure. The idea of measuring angles by the length of the arc was already in use by other mathematicians. For example al-Kashi (c. 1400) used so-called diameter parts as units where one diameter part was 1/60 radian and they also used sexagesimal subunits of the diameter part.

The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of theUniversity of St Andrews, vacillated between the terms radradial, and radian. In 1874, after a consultation with James Thomson, Muir adopted radian.


As stated, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π.

The radian is the standard unit of angular measure, used in many areas of mathematics. An angle's measurement in radians is numerically equal to the length of a corresponding arc of a unit circle, one radian is just under 57.3degrees (when the arc length is equal to the radius). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit. The SI unit of solid angle measurement is the steradian.

The radian is represented by the symbol rad (Unicode-encoded as U+33AD ㎭). An alternative symbol is the superscript letter c, for "circular measure"—but this is infrequently used.


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.

On the equinox, when the day and night is equal, you can divide the day and night into 180 degree segments. Each 180 degree segment is divided into 12 hours, which each hour is 15 degrees of sky. Some say this is because each of the constellations of the zodiac fill an hour of the sky as they rise above the horizon. 24 hours, 60 minutes, 60 seconds, they all work within the base 60 numeral system we've inherited.

Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled.


In 1936, a tablet was excavated some 200 miles from Babylon.  Here one should make the interjection that the Sumerians were first to make one of man's greatest inventions, namely, writing; through written communication, knowledge could be passed from one person to others, and from one generation to the next and future ones.  They impressed their cuneiform (wedge-shaped) script on soft clay tablets with a stylus, and the tablets were then hardened in the sun.  The mentioned tablet, whose translation was partially published only in 1950, is devoted to various geometrical figures, and states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60^2) (the Babylonians used the sexagesimal system, i.e., their base was 60 rather than 10).

The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day).  The tablet, therefore, gives ... Pi = 25/8 = 3.125.

So that's who gave us the 360 degrees in the circle.


And in parts of Europe they’ve used gradians, where you divide a circle into 400 pieces. The unit originated in France as the grade, along with the metric system, hence it is occasionally referred to a "metric degree." Due to confusion with existing grad(e) units of northern Europe, the name gon was later adopted, first in those regions, later as the international standard. In German, the unit was formerly also called Neugrad (new degree), likewise Nygrad in Swedish, Danish and Norwegian (also Gradian), and Nýgráða in Icelandic.

Although attempts at a general introduction were made, the unit was only adopted in some countries and for specialised areas such as surveying. The French artillery has used the grad for decades. 

Question: Why did the Sumerians used Sexagesimal Numerals?

Sumerians used sexagesimal numerals not only because the number 60 has many divisors or it is countable on the fingers of both hands but because 60 is the least common multiple of the number of fingers of both hands and the number of months in a year. Today both the Mesopotamian sexagesimal system is apt to be explained in terms of the 60-year conjunction cycle of Jupiter and Saturn.

"Chance favors the prepared mind." - Louis Pasteur