Radians are a unit that measure angle as the ratio of the angle’s arc-length over the radius of a circle. A full rotation in radians is equal to (tau) radians.

Note: This website uses the constant (tau) instead of (pi) as the default circle constant. The substitution can be used to translate between the two constants.

Usage

The radian system is used for measuring angles and as the unit of choice for the trigonometric functions. While the degree angle system is often used to introduce concepts, the radian system eventually becomes the preferred unit for measuring angles in math[1].

Definition

Radians are a unit that measure angle as the ratio of the angle’s arc-length over the radius of a circle. The (equivalent) symbol is used to indicate that negative angles or angles more than a full rotation are equivalent angle to an angle between and a full rotation. A full rotation is equal to (tau) radians.

Measuring Angles

Angles measured using radians are usually expressed using the circle constant (tau). Shown below are some examples of angles measured using radians. The variable (theta) is a variable commonly used for angles. By convention, angles in the coordinate plane are measured from the positive direction where the counter-clockwise rotation is positive.

Trigonometric Functions

The trigonometric functions and the unit circle are often where radians are introduced. Shown below are the plots of sine and cosine labeled in radians.