An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2π radians.

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results.

Most notably, results in analysis involving trigonometric functions are simple and elegant when the functions' arguments are expressed in radians.

Although the radian is a unit of measure, it is a dimensionless quantity.

2π radians equals one turn which is 360 degrees.

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