Radian is a way to measure angles very similar to degrees. It is a way to measure angles using their radius as a standard unit. A certain angle would be measured in the length of the arc divided by the radius. Since a circle’s circumference is 2(pi)r, the angle that’s equivalent to 360 degrees (full circle) is 2(pi)r/r = 2(pi), and 180 degrees =(pi) radians and so forth.
In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a “naturalness” that makes resulting formulae more elegant and simple. For example: “what is the area of a section of a circle with a radius of 2cm if the section angle is 2 radians?” and “what is the area of a section of a circle with a radius of 2cm if the section angle is 38 degrees?” are two similar questions with different units. The calculation of either question is the same, but the result of the radian question is much more pleasant. (pi)*2*2 * 2/2(pi) = 4cm(squared) is the answer for the radians question. (pi)*2*2 * 38/360 = 1.32645023151569047846200498405cm(squared) is the answer for the degrees question. Obviously, the result achieved from a question in radians is much more pleasant.
Radians also has the “unitless” quality, since it is simply a percentage of the circumference, with 2(pi) being 100%. This means that operations could be carried out using radians with out worrying about changing the units. For example, if one want to calculate the speed of an object from the distance it traveled and the time taken for this object to travel the distance, one would simply divide the distance by the time to get a number for speed. Analyzing the units reveal that distance (m) divided by time (s) equals speed (m/s) with the right units. Knowing that, if one would calculate a number using a function which require the use of radians, he/she would be assure by the fact that radians is unitless and would not change the units of the result causing need for further changes and calculations. The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically measured in radians per second (rad/s). One revolution per second is equal to 2 radians per second. Similarly, angular acceleration is often measured in radians per second per second (rad/s(squared)).
The radian is an important mathematical and scientific value, which is very commonly used for simplicity in calculation.